For Class 8, 9 & 10
Master the basics - and everything after gets easier
Concept-first questions with clear model answers in Physics, Chemistry, Maths and Biology, all NCERT-aligned. Start early, build the habit, and walk into your boards, NEET and JEE already ahead.
Interactive lessons
learn by playingDrag, slide and build - watch each concept come alive, then reveal the answer.
200 interactive lessons
Ohm's law
Class 10 Physics
Slide V & R, watch the bulb glow
Open →pH scale
Class 10 Chemistry
Slide across acids and bases
Open →Atomic number and mass number
Class 9 Chemistry
Add protons & neutrons, build shells
Open →Laws of reflection
Class 8 Physics
Change the angle, watch it bounce
Open →Volume of a sphere
Class 9 Maths
Grow the radius, see the volume
Open →Area of a trapezium
Class 8 Maths
Drag the sides, read the area
Open →Power of a lens
Class 10 Physics
Move the object, trace the rays
Open →Food chain and energy flow
Class 10 Biology
Follow the energy as it flows
Open →Speed
Class 8 Physics
Slide distance & time, watch the speed
Open →Density
Class 9 Physics
Pack mass into volume, float or sink
Open →Work done
Class 9 Physics
Push harder or farther, watch work grow
Open →Kinetic energy
Class 9 Physics
Speed it up - energy grows with the square
Open →Power of a lens
Class 10 Physics
Shorten the focal length, boost the power
Open →Mole concept
Class 9 Chemistry
Weigh out grams, count the moles
Open →Avogadro's number
Class 9 Chemistry
Add moles, count the particles
Open →Microscope magnification
Class 8 Biology
Grow the image, read the magnification
Open →Population density
Class 10 Biology
Add individuals, shrink the land, see crowding
Open →Simple interest
Class 8 Maths
Slide money, rate & time, watch interest
Open →Pythagoras theorem
Class 9 Maths
Stretch the two sides, get the hypotenuse
Open →Probability of an event
Class 10 Maths
Change the outcomes, watch the odds
Open →Newton's second law
Class 9 Physics
Push a mass, pick an acceleration
Open →Momentum
Class 9 Physics
Slide mass & velocity, build momentum
Open →Pressure
Class 8 Physics
Shrink the area, feel the pressure rise
Open →Weight
Class 9 Physics
Change the planet's gravity, watch your weight
Open →Refractive index
Class 10 Physics
Slow light in the medium, raise the index
Open →Resistors in series
Class 10 Physics
Add two resistors in a line
Open →Mass percentage of a solution
Class 9 Chemistry
Dissolve solute, read the strength
Open →Concentration of a solution
Class 9 Chemistry
Pack solute into less liquid
Open →Population change
Class 10 Biology
Balance births against deaths
Open →Compound microscope
Class 8 Biology
Combine eyepiece & objective lenses
Open →Area of a circle
Class 8 Maths
Grow the radius, watch the area square
Open →Volume of a cuboid
Class 8 Maths
Stretch length, breadth & height
Open →Electronic configuration and valency
Class 9 Chemistry
Slide the atomic number, build the atom
Open →Homologous series (alkanes)
Class 10 Chemistry
Add carbons, name the compound
Open →Mass number
Class 9 Chemistry
Add protons & neutrons, get the mass number
Open →Power
Class 9 Physics
More work in less time = more power
Open →Potential energy
Class 9 Physics
Lift a mass higher, store energy
Open →Wave speed
Class 9 Physics
Tune frequency & wavelength, set the speed
Open →Electric current
Class 10 Physics
Push charge per second, get the current
Open →Percentage
Class 8 Maths
Compare part to whole as a %
Open →Electron dot structure
Class 9 Chemistry
Draw valence electrons as dots
Open →Acceleration
Class 9 Physics
Speed up over time, find acceleration
Open →Distance, speed and time
Class 8 Physics
Set speed & time, cover the distance
Open →Frequency and time period
Class 9 Physics
Shorten the period, raise the frequency
Open →Heating effect of current
Class 10 Physics
Raise the current, watch heating soar
Open →Area of a triangle
Class 8 Maths
Set base & height, halve the rectangle
Open →Area of a rectangle
Class 8 Maths
Set length & breadth, fill the area
Open →Mean (average)
Class 9 Maths
Share the total equally across items
Open →Discount
Class 8 Maths
Slide price & % off, see the saving
Open →Heart rate
Class 10 Biology
Set heart rate & time, count the beats
Open →Resistors in parallel
Class 10 Physics
Wire two resistors side by side
Open →Electric charge
Class 10 Physics
Flow current over time, collect charge
Open →Electrical energy and units
Class 10 Physics
Run appliances, add up the units
Open →Kelvin temperature scale
Class 9 Chemistry
Slide Celsius, read the Kelvin
Open →Moles from number of particles
Class 9 Chemistry
Divide particles by Avogadro's number
Open →Ten percent law
Class 10 Biology
See 10% of energy reach the next level
Open →Area of a square
Class 8 Maths
Grow the side, square the area
Open →Volume of a cube
Class 8 Maths
Grow the edge, cube the volume
Open →Circumference of a circle
Class 8 Maths
Grow the radius, roll out the rim
Open →Surface area of a cube
Class 9 Maths
Grow the edge, cover six faces
Open →Potential difference
Class 10 Physics
Share work across charge, get volts
Open →Resistance from Ohm's law
Class 10 Physics
Divide voltage by current, get resistance
Open →Echo and SONAR
Class 9 Physics
Time the echo, find the distance
Open →Mass from moles
Class 9 Chemistry
Multiply moles by molar mass
Open →Breathing rate
Class 10 Biology
Set breathing rate & time
Open →Volume of a cylinder
Class 10 Maths
Set radius & height, fill the can
Open →Compound interest
Class 8 Maths
Compound money over years
Open →Profit and loss percentage
Class 8 Maths
Set cost & selling price, see profit %
Open →Perimeter of a rectangle
Class 8 Maths
Set length & breadth, walk the border
Open →Surface area of a sphere
Class 10 Maths
Grow the radius, wrap the ball
Open →Time period
Class 9 Physics
Raise the frequency, shrink the period
Open →Relative velocity
Class 9 Physics
Two objects approach - add their speeds
Open →Average velocity
Class 9 Physics
Average the start and end speeds
Open →Equations of motion (v = u + at)
Class 9 Physics
Accelerate from u for a time t
Open →Kelvin to Celsius
Class 9 Chemistry
Slide Kelvin, read the Celsius
Open →Population growth rate
Class 10 Biology
Balance births vs deaths per population
Open →Perimeter of a square
Class 8 Maths
Grow the side, walk four edges
Open →Perimeter of a triangle
Class 8 Maths
Add the three sides
Open →Area of a parallelogram
Class 8 Maths
Set base & height, slide the shape
Open →Area of a rhombus
Class 8 Maths
Set the two diagonals
Open →Equations of motion (distance)
Class 9 Physics
Start, accelerate, cover ground
Open →Joule's law of heating
Class 10 Physics
Raise current, resistance or time
Open →Electric power (P = VI)
Class 10 Physics
Multiply voltage by current
Open →Average atomic mass of isotopes
Class 9 Chemistry
Mix two isotopes by abundance
Open →Seed germination percentage
Class 9 Biology
Count sprouted seeds out of the total
Open →Volume of a cone
Class 9 Maths
Set radius & height, fill the cone
Open →Surface area of a cylinder
Class 9 Maths
Wrap the side and both ends
Open →Surface area of a cuboid
Class 9 Maths
Cover all six rectangular faces
Open →nth term of an AP
Class 10 Maths
Step from the first term by d
Open →Sum of an AP
Class 10 Maths
Add up the first n terms
Open →Focal length of a mirror
Class 10 Physics
Halve the radius to find the focus
Open →Speed of light in a medium
Class 10 Physics
Raise the index, slow the light
Open →Percentage purity
Class 9 Chemistry
Weigh the pure part of a sample
Open →Slope of a line
Class 10 Maths
Rise over run gives the steepness
Open →Percentage change
Class 8 Maths
Compare a new value to the old
Open →Volume of a hemisphere
Class 9 Maths
Grow the radius of half a ball
Open →Area of a sector
Class 10 Maths
Cut a slice of angle from a circle
Open →Length of an arc
Class 10 Maths
Measure the curved edge of a slice
Open →Slant height of a cone
Class 9 Maths
Combine radius & height for the slant
Open →Unit conversion (km/h to m/s)
Class 9 Physics
Convert km/h into m/s
Open →Equations of motion (v^2 = u^2 + 2as)
Class 9 Physics
Accelerate over a distance, find v
Open →Impulse
Class 9 Physics
Hit harder or longer, change momentum
Open →Wavelength
Class 9 Physics
Speed over frequency gives wavelength
Open →Oscillations
Class 9 Physics
Vibrate at a frequency for a time
Open →Cost of electricity
Class 10 Physics
Units times rate gives the bill
Open →Number of neutrons
Class 9 Chemistry
Take protons away from the mass number
Open →Curved surface area of a cone
Class 9 Maths
Wrap the slanted side of a cone
Open →Total surface area of a cone
Class 9 Maths
Add the base circle to the cone's side
Open →Curved surface area of a hemisphere
Class 9 Maths
Cover the dome of a hemisphere
Open →Diagonal of a square
Class 9 Maths
Cross a square corner to corner
Open →Unit conversion (m/s to km/h)
Class 9 Physics
Convert m/s into km/h
Open →Distance from velocities
Class 9 Physics
From two speeds, find the distance
Open →Diagonal of a rectangle
Class 9 Maths
Cross a rectangle corner to corner
Open →Diagonal of a cuboid
Class 9 Maths
The longest rod that fits in a box
Open →Area by Heron's formula
Class 9 Maths
Area from just the three sides
Open →Interior angle sum of a polygon
Class 8 Maths
Add up a polygon's inside angles
Open →Exterior angle of a regular polygon
Class 8 Maths
Share 360 among a polygon's corners
Open →Number of diagonals of a polygon
Class 8 Maths
Count the diagonals of a polygon
Open →Discriminant
Class 10 Maths
Test how many roots a quadratic has
Open →Sum of roots
Class 10 Maths
Sum of a quadratic's roots
Open →Punnett square (monohybrid cross)
Class 10 Biology
Cross two parents, predict the offspring
Open →Balancing chemical equations
Class 10 Chemistry
Slide coefficients until atoms balance
Open →Writing chemical formulae (valency)
Class 9 Chemistry
Criss-cross valencies into a formula
Open →Current from power
Class 10 Physics
Divide power by voltage for current
Open →Power (P = V^2 / R)
Class 10 Physics
Voltage squared over resistance
Open →Sine ratio
Class 10 Maths
Opposite over hypotenuse
Open →Cosine ratio
Class 10 Maths
Adjacent over hypotenuse
Open →Tangent ratio
Class 10 Maths
Opposite over adjacent
Open →Area of an equilateral triangle
Class 9 Maths
Area of an equilateral triangle
Open →Curved surface area of a cylinder
Class 9 Maths
Wrap only the curved side
Open →Loss percentage
Class 8 Maths
Sell below cost, find the loss %
Open →Amount with simple interest
Class 8 Maths
Principal plus its simple interest
Open →Distance formula
Class 10 Maths
Straight distance between two points
Open →States of matter
Class 9 Chemistry
Heat particles solid → liquid → gas
Open →Parts of a plant cell
Class 8 Biology
Tap a cell part to see its job
Open →Diagonal of a cube
Class 9 Maths
Longest diagonal through a cube
Open →Total surface area of a hemisphere
Class 9 Maths
Dome plus its flat circle
Open →Sum of first n natural numbers
Class 10 Maths
Add 1 + 2 + ... + n instantly
Open →Range of data
Class 9 Maths
Spread from smallest to largest
Open →Class mark
Class 9 Maths
Midpoint of a class interval
Open →Selling price from profit percent
Class 8 Maths
Mark up cost by a profit %
Open →Perimeter of a sector
Class 10 Maths
Two radii plus the curved arc
Open →Circumference from diameter
Class 8 Maths
Circumference straight from diameter
Open →Power (P = F x v)
Class 9 Physics
Force times velocity gives power
Open →Percentage of a number
Class 8 Maths
Find a percentage of a number
Open →Series and parallel circuits
Class 10 Physics
Break a bulb in series vs parallel
Open →Symbols of elements
Class 9 Chemistry
Match each element to its symbol
Open →Free fall (velocity)
Class 9 Physics
Drop from a height, hit this speed
Open →Free fall (time)
Class 9 Physics
How long a drop takes
Open →Free fall (distance)
Class 9 Physics
Distance fallen in a given time
Open →Complement of an event
Class 10 Maths
Chance an event does NOT happen
Open →Product of roots
Class 10 Maths
Product of a quadratic's roots
Open →Exterior angle theorem
Class 9 Maths
Exterior angle = sum of remote interiors
Open →Complementary angles
Class 10 Maths
What adds to 90 degrees
Open →Supplementary angles
Class 9 Maths
What adds to 180 degrees
Open →Perimeter of a semicircle
Class 10 Maths
Curved half plus the diameter
Open →Area of a semicircle
Class 10 Maths
Half the area of a circle
Open →Turning effect (moments)
Class 9 Physics
Balance the see-saw with moments
Open →Reflex arc
Class 10 Biology
Step through a reflex, stimulus to action
Open →Buoyant force (upthrust)
Class 9 Physics
Displace liquid, feel the upthrust
Open →Relative density
Class 9 Physics
Compare a density to water's
Open →Power in lifting a load
Class 9 Physics
Lift a load, faster needs more power
Open →Cosecant ratio
Class 10 Maths
Hypotenuse over opposite
Open →Secant ratio
Class 10 Maths
Hypotenuse over adjacent
Open →Cotangent ratio
Class 10 Maths
Adjacent over opposite
Open →Height from angle of elevation
Class 10 Maths
Height from an angle of elevation
Open →Area of a quadrant
Class 10 Maths
A quarter of a circle's area
Open →Interior angle of a regular polygon
Class 8 Maths
One inside angle of a regular polygon
Open →Sum of first n odd numbers
Class 10 Maths
Add the first n odd numbers
Open →Sum of first n even numbers
Class 10 Maths
Add the first n even numbers
Open →Quadratic formula (a root)
Class 10 Maths
Larger root of a quadratic
Open →LCM from HCF
Class 10 Maths
LCM from the product and HCF
Open →Depreciation
Class 8 Maths
Value drops by a % each year
Open →Cost price from selling price
Class 8 Maths
Work back to the cost price
Open →Downstream speed
Class 8 Maths
Row with the current
Open →Upstream speed
Class 8 Maths
Row against the current
Open →Average speed for a round trip
Class 8 Maths
Average speed there and back
Open →Sales tax / GST
Class 8 Maths
Tax added on a price
Open →Area of a ring (annulus)
Class 10 Maths
Area of a ring between two circles
Open →Edge of a cube from volume
Class 9 Maths
Edge back from the volume
Open →Radius from area
Class 10 Maths
Radius back from a circle's area
Open →Side from area of a square
Class 8 Maths
Side back from a square's area
Open →Height of a triangle from area
Class 9 Maths
Height back from area and base
Open →Rate from simple interest
Class 8 Maths
Rate back from the interest
Open →Time from simple interest
Class 8 Maths
Time back from the interest
Open →Principal from simple interest
Class 8 Maths
Principal back from the interest
Open →Mean proportional
Class 10 Maths
Geometric mean of two numbers
Open →Fourth proportional
Class 8 Maths
Complete the proportion a : b = c : ?
Open →Marked price from selling price
Class 8 Maths
Marked price back from the sale price
Open →Chambers of the human heart
Class 10 Biology
Tap a heart chamber to see its job
Open →Equation of a line (y = mx + c)
Class 9 Maths
Read y off a straight line
Open →Average term of an AP
Class 10 Maths
Average of first and last term
Open →Number of terms in an AP
Class 10 Maths
How many terms in an AP
Open →Midpoint of two points
Class 10 Maths
x-coordinate of a midpoint
Open →Empirical mode
Class 10 Maths
Estimate the mode from mean & median
Open →Length of a shadow
Class 10 Maths
Shadow from height and sun angle
Open →Train crossing a pole
Class 8 Maths
Speed to cross a pole
Open →Time and work
Class 8 Maths
More workers, fewer days
Open →Dividing in a ratio
Class 8 Maths
Split a total in a ratio
Open →Unitary method
Class 8 Maths
Cost of a single item
Open →Your progress — Foundation - Class 8 to 10
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Showing 12 questions in Mathematics for Class 10. Tap a card to reveal the answer.
MathematicsIntroduction to TrigonometrymediumIf sin(theta) = 3/5, find cos(theta) and tan(theta), where theta is acute.
Reveal answer ↓
What it is
The identity sin^2 + cos^2 = 1 lets you find one trig ratio from another for the same angle.
Answer
Using the identity sin^2(theta) + cos^2(theta) = 1, cos^2(theta) = 1 - (3/5)^2 = 1 - 9/25 = 16/25, so cos(theta) = 4/5 (positive since theta is acute). Then tan(theta) = sin/cos = (3/5) / (4/5) = 3/4.
sin^2(theta) + cos^2(theta) = 1 ; tan = sin/cos
- •sin^2 + cos^2 = 1
- •cos(theta) = 4/5
- •tan(theta) = sin/cos = 3/4
Why learn this
Trig ratios drive navigation, physics waves, construction and computer graphics.
💡 Memory trick
Think of the 3-4-5 triangle: sin 3/5 -> cos 4/5 -> tan 3/4.
MathematicsIntroduction to TrigonometrymediumEvaluate 2 tan^2(45 degrees) + cos^2(30 degrees) - sin^2(60 degrees).
Reveal answer ↓
What it is
The standard angles 0, 30, 45, 60 and 90 degrees have fixed trig values worth memorising.
Answer
Use standard values: tan(45 deg) = 1, cos(30 deg) = sqrt(3)/2, sin(60 deg) = sqrt(3)/2. So the expression = 2(1)^2 + (sqrt(3)/2)^2 - (sqrt(3)/2)^2 = 2 + 3/4 - 3/4 = 2.
tan45 = 1 ; cos30 = sqrt(3)/2 ; sin60 = sqrt(3)/2
- •tan45 = 1, cos30 = sin60 = sqrt(3)/2
- •2(1) + 3/4 - 3/4
- •= 2
Why learn this
They appear in nearly every physics and maths problem - knowing them saves huge time.
💡 Memory trick
For sin, use sqrt(0..4)/2 across 0,30,45,60,90; cos is the same list reversed.
MathematicsIntroduction to TrigonometrymediumIn a right triangle the opposite side is 3 and the hypotenuse is 5. Find sin of the angle. Slide to explore.
Reveal answer ↓
What it is
In a right triangle, the sine of an angle is the opposite side divided by the hypotenuse.
Answer
sin(theta) = opposite / hypotenuse = 3 / 5 = 0.6. The sine of an acute angle is always between 0 and 1, because the opposite side is never longer than the hypotenuse.
sin(theta) = opposite / hypotenuse
- •sin = opposite / hypotenuse (SOH)
- •Always between 0 and 1
- •3-4-5 triangle gives sin = 0.6
Why learn this
Sine links an angle to side lengths - the heart of trigonometry.
💡 Memory trick
SOH: Sine = Opposite / Hypotenuse.
MathematicsIntroduction to TrigonometrymediumIn a right triangle the adjacent side is 4 and the hypotenuse is 5. Find cos of the angle. Slide to explore.
Reveal answer ↓
What it is
In a right triangle, the cosine of an angle is the adjacent side divided by the hypotenuse.
Answer
cos(theta) = adjacent / hypotenuse = 4 / 5 = 0.8. Like sine, it lies between 0 and 1, and the two obey the identity sin^2(theta) + cos^2(theta) = 1.
cos(theta) = adjacent / hypotenuse
- •cos = adjacent / hypotenuse (CAH)
- •Always between 0 and 1
- •sin^2 + cos^2 = 1
Why learn this
Cosine pairs with sine to resolve forces, waves and vectors.
💡 Memory trick
CAH: Cosine = Adjacent / Hypotenuse.
MathematicsIntroduction to TrigonometrymediumIn a right triangle the opposite side is 3 and the adjacent side is 4. Find tan of the angle. Slide to explore.
Reveal answer ↓
What it is
In a right triangle, the tangent of an angle is the opposite side divided by the adjacent side.
Answer
tan(theta) = opposite / adjacent = 3 / 4 = 0.75. It also equals sin(theta) / cos(theta), and it grows without limit as the angle approaches 90 degrees.
tan(theta) = opposite / adjacent
- •tan = opposite / adjacent (TOA)
- •tan = sin / cos
- •Grows large near 90 degrees
Why learn this
Tangent gives the slope of a line and heights from angles of elevation.
💡 Memory trick
TOA: Tangent = Opposite / Adjacent.
MathematicsIntroduction to TrigonometryeasyWhat is the complement of a 30 degree angle? Slide the angle to explore.
Reveal answer ↓
What it is
Two angles are complementary when they add up to 90 degrees.
Answer
Complement = 90 - angle = 90 - 30 = 60 degrees. Complementary angles add to a right angle, which is why the sine of an angle equals the cosine of its complement: sin(30) = cos(60).
complement = 90 - angle
- •Complement = 90 - angle
- •The two add to 90 degrees
- •sin(theta) = cos(90 - theta)
Why learn this
Complementary angles link sine and cosine: sin(theta) = cos(90 - theta).
💡 Memory trick
Complement = 90 - angle.
MathematicsIntroduction to TrigonometrymediumIn a right triangle the hypotenuse is 5 and the opposite side is 3. Find cosec of the angle. Slide to explore.
Reveal answer ↓
What it is
The cosecant of an angle is the hypotenuse divided by the opposite side, the reciprocal of sine.
Answer
cosec(theta) = hypotenuse / opposite = 5 / 3 = 1.667. It is the reciprocal of the sine ratio, so cosec(theta) = 1 / sin(theta).
cosec(theta) = hypotenuse / opposite
- •cosec = hypotenuse / opposite
- •cosec = 1 / sin
- •A reciprocal ratio
Why learn this
The reciprocal ratios complete the trigonometric toolkit.
💡 Memory trick
cosec = hypotenuse / opposite = 1 / sin.
MathematicsIntroduction to TrigonometrymediumIn a right triangle the hypotenuse is 5 and the adjacent side is 4. Find sec of the angle. Slide to explore.
Reveal answer ↓
What it is
The secant of an angle is the hypotenuse divided by the adjacent side, the reciprocal of cosine.
Answer
sec(theta) = hypotenuse / adjacent = 5 / 4 = 1.25. It is the reciprocal of the cosine ratio, so sec(theta) = 1 / cos(theta).
sec(theta) = hypotenuse / adjacent
- •sec = hypotenuse / adjacent
- •sec = 1 / cos
- •A reciprocal ratio
Why learn this
It appears whenever the cosine sits in a denominator.
💡 Memory trick
sec = hypotenuse / adjacent = 1 / cos.
MathematicsIntroduction to TrigonometrymediumIn a right triangle the adjacent side is 4 and the opposite side is 3. Find cot of the angle. Slide to explore.
Reveal answer ↓
What it is
The cotangent of an angle is the adjacent side divided by the opposite side, the reciprocal of tangent.
Answer
cot(theta) = adjacent / opposite = 4 / 3 = 1.333. It is the reciprocal of the tangent ratio, so cot(theta) = 1 / tan(theta).
cot(theta) = adjacent / opposite
- •cot = adjacent / opposite
- •cot = 1 / tan
- •A reciprocal ratio
Why learn this
It completes the six trigonometric ratios.
💡 Memory trick
cot = adjacent / opposite = 1 / tan.
MathematicsIntroduction to TrigonometrymediumDefine sin, cos and tan of an acute angle in a right-angled triangle.
Reveal answer ↓
What it is
In a right triangle, the trigonometric ratios sine, cosine and tangent relate an angle to pairs of its sides.
Answer
In a right-angled triangle, for an acute angle A: the sine of A is the ratio of the side opposite to A to the hypotenuse (sin A = opposite/hypotenuse); the cosine of A is the ratio of the side adjacent to A to the hypotenuse (cos A = adjacent/hypotenuse); and the tangent of A is the ratio of the side opposite to A to the side adjacent to A (tan A = opposite/adjacent). Also, tan A = sin A / cos A. The reciprocals are cosec A, sec A and cot A.
sin = opp/hyp ; cos = adj/hyp ; tan = opp/adj
- •sin A = opposite / hypotenuse
- •cos A = adjacent / hypotenuse
- •tan A = opposite / adjacent
- •tan A = sin A / cos A
Why learn this
They connect angles to side lengths and are the basis of all trigonometry.
💡 Memory trick
SOH-CAH-TOA: sin = Opp/Hyp, cos = Adj/Hyp, tan = Opp/Adj.
MathematicsIntroduction to TrigonometrymediumEvaluate sin 30 + cos 60 and tan 45.
Reveal answer ↓
What it is
The trig ratios of 0, 30, 45, 60 and 90 degrees have fixed standard values.
Answer
Using the standard values: sin 30 = 1/2 and cos 60 = 1/2, so sin 30 + cos 60 = 1/2 + 1/2 = 1. Also tan 45 = 1. So sin 30 + cos 60 = 1 and tan 45 = 1.
sin 30 = 1/2 ; cos 60 = 1/2 ; tan 45 = 1
- •sin 30 = 1/2, cos 60 = 1/2
- •sin 30 + cos 60 = 1
- •tan 45 = 1
- •Learn the standard-angle table
Why learn this
They are used constantly in solving triangles and applications.
💡 Memory trick
sin: 0, 1/2, 1/sqrt2, sqrt3/2, 1 for 0, 30, 45, 60, 90. cos is the reverse.
MathematicsIntroduction to TrigonometrymediumState the three trigonometric identities. If sin A = 3/5, find cos A.
Reveal answer ↓
What it is
The fundamental trigonometric identity is sin^2 A + cos^2 A = 1.
Answer
The three fundamental trigonometric identities are: sin^2 A + cos^2 A = 1; 1 + tan^2 A = sec^2 A; and 1 + cot^2 A = cosec^2 A. Given sin A = 3/5, use sin^2 A + cos^2 A = 1: cos^2 A = 1 - sin^2 A = 1 - (3/5)^2 = 1 - 9/25 = 16/25. So cos A = sqrt(16/25) = 4/5 (taking the positive value for an acute angle).
sin^2 A + cos^2 A = 1
- •sin^2 A + cos^2 A = 1
- •1 + tan^2 A = sec^2 A
- •1 + cot^2 A = cosec^2 A
- •cos A = 4/5 when sin A = 3/5
Why learn this
It lets us find one ratio from another and simplify expressions.
💡 Memory trick
sin^2 + cos^2 = 1 always. The other two: 1 + tan^2 = sec^2 ; 1 + cot^2 = cosec^2.
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