Degrees and radians
One radian is the angle at the centre when the arc is as long as the radius (step 1 of the 3D). The whole circle has arc 2πr, so it is 2π radians.
- π rad = 180°, so 1 rad = 180°/π ≈ 57°16′.
- Degrees → radians: multiply by π/180. Radians → degrees: multiply by 180/π.
- Arc length l = r θ (θ in radians).
- 1° = 60′ (minutes), 1′ = 60″ (seconds).
Unit-circle definitions and sin²x + cos²x = 1
Draw a circle of radius 1 with centre at the origin. Turn a radius by angle x from the positive x-axis. It ends at P(a, b). We define cos x = a and sin x = b.
This works for every real x, even more than 360° or negative. Then tan x = sin x / cos x (cos x ≠ 0), cot x = cos x / sin x, sec x = 1/cos x, cosec x = 1/sin x.
P is at distance 1 from the centre, so by Pythagoras a² + b² = 1, which is sin²x + cos²x = 1. Divide by cos²x: 1 + tan²x = sec²x. Divide by sin²x: 1 + cot²x = cosec²x.
Signs, domain, range and graphs
Quadrant signs (step 3): I all +, II sin and cosec +, III tan and cot +, IV cos and sec +.
- sin x, cos x: domain R, range [−1, 1], period 2π.
- tan x: domain R − {(2n + 1)π/2}, range R, period π.
- cot x: domain R − {nπ}, range R.
- sec x: domain R − {(2n + 1)π/2}, range (−∞, −1] ∪ [1, ∞). cosec x: domain R − {nπ}, same range.
- sin x = 0 at x = nπ; cos x = 0 at x = (2n + 1)π/2.
- sin(−x) = −sin x (odd), cos(−x) = cos x (even).
Step 4 draws y = sin x from the circle: it starts at 0, peaks at 1 at π/2, returns to 0 at π, drops to −1 at 3π/2. cos x is the same wave shifted left by π/2.
Sum and difference formulas
- cos(A + B) = cos A cos B − sin A sin B
- cos(A − B) = cos A cos B + sin A sin B
- sin(A + B) = sin A cos B + cos A sin B
- sin(A − B) = sin A cos B − cos A sin B
- tan(A + B) = (tan A + tan B)/(1 − tan A tan B)
- tan(A − B) = (tan A − tan B)/(1 + tan A tan B)
- cot(A + B) = (cot A cot B − 1)/(cot B + cot A)
Special cases: cos(π/2 − x) = sin x, sin(π/2 − x) = cos x, sin(π − x) = sin x, cos(π − x) = −cos x, sin(π + x) = −sin x, cos(2π − x) = cos x.
Idea of the proof (in our own steps): put P at angle A + B and Q at angle 0; then put R at angle A and S at angle −B. The chord PQ and the chord RS subtend the same angle A + B, so they are equal. Write both with the distance formula, simplify using sin² + cos² = 1, and cos(A + B) comes out. The rest follow by replacing B with −B or A with π/2 − A.
Double, triple angle and sum-to-product
Double angle (put B = A)
- sin 2x = 2 sin x cos x = 2 tan x/(1 + tan²x)
- cos 2x = cos²x − sin²x = 2cos²x − 1 = 1 − 2sin²x = (1 − tan²x)/(1 + tan²x)
- tan 2x = 2 tan x/(1 − tan²x)
Triple angle
- sin 3x = 3 sin x − 4 sin³x
- cos 3x = 4 cos³x − 3 cos x
- tan 3x = (3 tan x − tan³x)/(1 − 3 tan²x)
Sum to product
- cos x + cos y = 2 cos((x + y)/2) cos((x − y)/2)
- cos x − cos y = −2 sin((x + y)/2) sin((x − y)/2)
- sin x + sin y = 2 sin((x + y)/2) cos((x − y)/2)
- sin x − sin y = 2 cos((x + y)/2) sin((x − y)/2)
And back: 2 sin x cos y = sin(x + y) + sin(x − y), 2 cos x cos y = cos(x + y) + cos(x − y), −2 sin x sin y = cos(x + y) − cos(x − y).
Try it
At home: tie a thread to a bottle cap, stretch it as a radius, then lay the same length of thread along the rim of a round plate. Mark the angle it makes at the centre: that is 1 radian. How many such threads fit round the rim? (A little more than 6, because 2π ≈ 6.28.)
In the 3D: in step 5 set A = 45°, B = 30° and check the readout gives sin 75° ≈ 0.966. In free play, predict the sign of cos 200° before you slide.
Key formulas and definitions
- π rad = 180°, l = rθ
- sin²x + cos²x = 1, 1 + tan²x = sec²x, 1 + cot²x = cosec²x
- sin(A ± B) = sin A cos B ± cos A sin B
- cos(A ± B) = cos A cos B ∓ sin A sin B
- tan(A ± B) = (tan A ± tan B)/(1 ∓ tan A tan B)
- sin 2x = 2 sin x cos x, cos 2x = 1 − 2 sin²x, tan 2x = 2tan x/(1 − tan²x)
- sin 3x = 3 sin x − 4 sin³x, cos 3x = 4 cos³x − 3 cos x
- cos x + cos y = 2 cos((x+y)/2) cos((x−y)/2)
Worked examples
1. Convert 40°20′ into radians.
40°20′ = 40⅓° = 121/3°. Radians = (121/3) × π/180 = 121π/540.
2. Convert 6 radians into degrees (π ≈ 22/7).
6 × 180/π = 1080 × 7/22 ≈ 343.64° ≈ 343°38′.
3. A wheel of radius 35 cm turns through 72°. How far does a point on the rim move?
θ = 72 × π/180 = 2π/5. l = rθ = 35 × 2π/5 = 14π ≈ 44 cm.
4. If cos x = −3/5 and x is in quadrant III, find the other five ratios.
sin²x = 1 − 9/25 = 16/25. In III sin is negative: sin x = −4/5. tan x = (−4/5)/(−3/5) = 4/3, cot x = 3/4, sec x = −5/3, cosec x = −5/4.
5. Find sin 765° and cos(−1710°).
765° = 2 × 360° + 45°, so sin 765° = sin 45° = 1/√2. −1710° = −5 × 360° + 90°, so cos(−1710°) = cos 90° = 0.
6. Find sin 75° (the worked example in step 5).
sin(45° + 30°) = sin 45° cos 30° + cos 45° sin 30° = (1/√2)(√3/2) + (1/√2)(1/2) = (√3 + 1)/(2√2) ≈ 0.966.
7. Find tan 15°.
tan(45° − 30°) = (1 − 1/√3)/(1 + 1/√3) = (√3 − 1)/(√3 + 1) = (√3 − 1)²/2 = (4 − 2√3)/2 = 2 − √3.
8. Prove (sin 5x + sin 3x)/(cos 5x + cos 3x) = tan 4x.
Numerator = 2 sin 4x cos x; denominator = 2 cos 4x cos x (sum to product). Ratio = sin 4x / cos 4x = tan 4x.
9. If sin x = 1/3, find sin 3x and cos 2x.
sin 3x = 3(1/3) − 4(1/27) = 1 − 4/27 = 23/27. cos 2x = 1 − 2(1/9) = 7/9.
Common mistakes
- Using degrees in l = rθ. θ must be in radians.
- Writing sin(A + B) = sin A + sin B. The sum formula has four terms: sin A cos B + cos A sin B.
- Forgetting the quadrant sign. After using sin²x + cos²x = 1, choose + or − from the quadrant (step 3).
- Thinking sin²x means sin(x²). sin²x = (sin x)².