Shunya / Trigonometry & Height-Distance

Shunya · Maths lab

Trigonometry & heights and distances

Spin a point round the unit circle and read sin, cos and tan straight off it. Watch the circle draw the sine wave, see why the ratios of a right triangle never change with its size, then use them to find the height of a tower, the distance to a boat or the length of a kite string by dragging the scene yourself.

Lessons

Pick your class, follow the steps in the lab, and check what you see against the working under the graph.

Class 10 — Trigonometric ratios of an acute angle

Concept: in a right triangle, for an acute angle $\theta$ : $\sin\theta = \tfrac{\text{opposite}}{\text{hypotenuse}}$ , $\cos\theta = \tfrac{\text{adjacent}}{\text{hypotenuse}}$ , $\tan\theta = \tfrac{\text{opposite}}{\text{adjacent}}$ .

Open Right triangle and read the three sides and the three ratios.
Drag the Size slider. The sides change but sin, cos and tan do not: they depend only on the angle.
Press 30° – 60° – 90° and 45° – 45° – 90° and read the exact values (1/2, √3/2, 1/√2, …).
Press 3 – 4 – 5: sin θ = 3/5, cos θ = 4/5, tan θ = 3/4.
Check the complementary angles: sin(90° − θ) = cos θ.

Takeaway: a trigonometric ratio is a fixed number for a given angle, whatever the size of the triangle.

Class 10 — Trigonometric identities

Concept: $\sin^2\theta + \cos^2\theta = 1$ and $1 + \tan^2\theta = \sec^2\theta$ hold for every angle. They come from Pythagoras.

In Right triangle, read the working: sin²θ + cos²θ = 1 with the numbers filled in.
Change the angle and confirm it stays 1 each time, even for a very small or very large triangle.
Open Unit circle. The point P = (cos θ, sin θ) is always on the circle x² + y² = 1, which is the identity itself.

Takeaway: the identity is just the Pythagoras theorem on a triangle with hypotenuse 1.

Class 10 — Heights and distances

Concept: the angle of elevation is measured up from the horizontal to the line of sight; the angle of depression is measured down. Draw the right triangle and pick the ratio that links the side you know to the one you want.

Open Heights & distances, choose Tower, and drag the observer. Watch θ and tan θ = H ÷ d.
Press Challenge: match the angle and move the observer until θ hits the target, then check d = H ÷ tan θ.
Choose Cliff & boat: the angle of depression equals the boat's angle of elevation (alternate angles).
Choose Kite: the string is the hypotenuse, so sin θ = height ÷ string.
Choose Two points: from two angles of elevation and the distance between the points, the height follows without measuring the distance to the tower.
Open Word problems and solve a few. Use the hint if you get stuck.

Takeaway: every height-and-distance problem is a right triangle with one angle and one side known.

Class 11 — The unit circle, radians and signs

Concept: on the unit circle, the point at angle θ is $(\cos\theta, \sin\theta)$ . Angles can be measured in radians: $180^\circ = \pi$ radians. The signs of the ratios depend on the quadrant: All, Sin, Tan, Cos are positive in quadrants I, II, III, IV.

Open Unit circle and drag the point round. Watch sin θ (vertical) and cos θ (horizontal) change sign.
Switch to Radians and press 30°, 45°, 60°, 90°. Read π/6, π/4, π/3, π/2.
Go to 150°. The reference angle is 180° − 150° = 30°, so sin 150° = sin 30° = 1/2 but cos 150° = −√3/2.
Try 90° and 270°: tan is undefined there because cos = 0.

Takeaway: a ratio's size comes from the reference angle and its sign from the quadrant.

Class 11 — Graphs of sin, cos and tan

Concept: plotting the height of the point against its angle gives the sine curve. The curve repeats every $2\pi$ (tan every $\pi$ ). For $y = A\sin(B(x - C)) + D$ the amplitude is |A|, the period is $2\pi/|B|$ , the phase shift is C and the midline is y = D.

Open Graphs and drag the point on the circle. Watch the marker travel along the wave.
Switch to cos and then tan. Where are the asymptotes of tan?
Tick show the transformed curve and press 2 sin x: the amplitude doubles.
Press sin 2x: the period halves to π. Press sin(x − 60°): the curve slides right.

Takeaway: A stretches it up and down, B squeezes it sideways, C slides it and D lifts it.

Class 11 & 12 — Reciprocal and inverse trigonometric graphs

Concept: cosec, sec and cot are the reciprocals of sin, cos and tan; the inverse functions undo them, but only on a restricted stretch called the principal value branch. Open Typed trig graphs, where any expression in x (in radians) can be graphed and analysed.

Press Compare a pair → sin x & cosec x. cosec x = 1 ÷ sin x has a vertical asymptote at every x = kπ (where sin x = 0). Since |sin x| ≤ 1, |cosec x| ≥ 1: the range is (−∞, −1] ∪ [1, ∞), and the two curves touch where sin x = ±1. The same works for sec x and cos x (asymptotes at π/2 + kπ), and tan x with cot x.
Press sin⁻¹ x: domain [−1, 1], range [−π/2, π/2]. Then cos⁻¹ x: range [0, π]. Press sin⁻¹x + cos⁻¹x and see the flat line at π/2 = 1.5708.
tan⁻¹ x is defined for every x and flattens out at y = ±π/2 (horizontal asymptotes). sec⁻¹ x = cos⁻¹(1 ÷ x) has domain (−∞, −1] ∪ [1, ∞) and range [0, π/2) ∪ (π/2, π].
The classic trap: press sin⁻¹(sin x). It is not x. It is x only for −π/2 ≤ x ≤ π/2 and then zig-zags with period 2π. For example sin⁻¹(sin 2π/3) = π/3, not 2π/3. Compare with sin(sin⁻¹ x), which is x but only on [−1, 1].
Harmonic form: type sqrt(3)*sin(x)-cos(x). The working finds R = √(3 + 1) = 2 and α = −π/6, so it equals 2 sin(x − π/6) with range [−2, 2]. Likewise sin x + cos x = √2 sin(x + π/4), range [−1.414, 1.414].

Takeaway: reciprocals have asymptotes where the original is 0; inverse functions are defined by a principal-value branch, and f⁻¹(f(x)) = x only on that branch.

Class 11 — Trigonometric equations and their general solutions

Concept: a trigonometric equation has infinitely many solutions because the functions repeat. The general solution lists them all with an integer n: reduce the equation to sin x = sin α, cos x = cos α or tan x = tan α, where α is the principal value.

Open Trig equations, press sin x = 1/2. The dashed line y = 1/2 crosses the curve at x = π/6 (30°) and 5π/6 (150°) in [0, 2π]. With α = π/6 the general solution is x = nπ + (−1)ⁿ α: for even n it gives π/6 + 2mπ, for odd n it gives π − π/6 = 5π/6 + 2mπ.
cos x = −1/2: α = cos⁻¹(−1/2) = 2π/3, and x = 2nπ ± 2π/3, which gives 2π/3 and 4π/3 in [0, 2π]. tan x = √3: α = π/3, x = nπ + π/3: one family, because tan repeats every π (π/3 and 4π/3 in [0, 2π]).
sin x = 1 touches the line only at the top: one solution, π/2, not two. A value outside [−1, 1], for instance k = 1.2, has no solution at all.
sin 2x = 1/2: put t = 2x, so t = mπ + (−1)^m π/6 and x = t ÷ 2. In [0, 2π] there are 4 solutions: π/12, 5π/12, 13π/12, 17π/12 (15°, 75°, 195°, 255°). Multiple angles give 2n solutions per period.
a sin x + b cos x = c. For √3 sin x + cos x = 1 write R = √(3 + 1) = 2 and tan φ = 1/√3, φ = π/6, so 2 sin(x + π/6) = 1, sin(x + π/6) = 1/2 and x = nπ + (−1)ⁿ π/6 − π/6, giving 0, 2π/3 and 2π in [0, 2π]. No solution exists if |c| > R.
Quadratic in sin x. 2 sin²x − sin x − 1 = 0 factorises as (2 sin x + 1)(sin x − 1) = 0, so sin x = 1 or −1/2: π/2, 7π/6, 11π/6. Always reject roots outside [−1, 1]. For 2 cos²x + cos x − 1 = 0 the roots are cos x = 1/2 and −1: π/3, π, 5π/3.

Takeaway: sin x = sin α ⇒ x = nπ + (−1)ⁿ α; cos x = cos α ⇒ x = 2nπ ± α; tan x = tan α ⇒ x = nπ + α. Check each root against the allowed range.

Class 11 — Compound angles, multiple angles and trigonometric identities

Concept: sin(A ± B) = sin A cos B ± cos A sin B, cos(A ± B) = cos A cos B ∓ sin A sin B and tan(A ± B) = (tan A ± tan B) ÷ (1 ∓ tan A tan B). Every other identity is built from these.

Open Compound angles & identities with A = 75° and B = 30°: sin 105° = 0.9659, cos 105° = −0.2588, tan 105° = −3.7321, and the table shows that the formula and the direct value agree. The exact values for 105° are sin = (√6 + √2) ÷ 4, cos = −(√6 − √2) ÷ 4, tan = −(2 + √3).
Press 75° = 45° + 30°: sin 75° = (1/√2)(√3/2) + (1/√2)(1/2) = (√6 + √2) ÷ 4 = 0.9659. Press 15° = 45° − 30°: sin 15° = (√6 − √2) ÷ 4 = 0.2588.
Double angle: put B = A. sin 2A = 2 sin A cos A, cos 2A = cos²A − sin²A = 2 cos²A − 1 = 1 − 2 sin²A, tan 2A = 2 tan A ÷ (1 − tan²A). Reading the cos 2A forms backwards gives the half-angle formulas sin²(A/2) = (1 − cos A) ÷ 2 and cos²(A/2) = (1 + cos A) ÷ 2 (the sign comes from the quadrant of A/2).
Sum to product: sin A + sin B = 2 sin((A + B)/2) cos((A − B)/2); the table checks it numerically (1.4659 on both sides for 75° and 30°).
Switch to Check an identity: choose sin 3x = 3 sin x − 4 sin³x and the two curves lie on top of each other (the largest gap is 0). Now choose sin 2x = 2 sin x (false!): the curves separate, the largest gap is 2.598, and x = 90° is the quick counter-example: sin 180° = 0 but 2 sin 90° = 2. One counter-example disproves a claim; to prove one, transform one side into the other.

Takeaway: sin and cos do not distribute over sums; use the compound-angle formulas, and test a claimed identity with numbers before trying to prove it.

The maths behind it

What each mode covers

Unit circle — drag a point round the circle for live sin, cos, tan, cosec, sec and cot, the exact values at the special angles, radians as multiples of π, the quadrant and its sign, and the reference angle. Class 11 trigonometric functions.

Graphs — the circle draws the sin, cos or tan curve as the point moves, with period, amplitude, phase shift and midline for

y = A f(B(x - C)) + D

. Class 11 graphs of trigonometric functions.

Right triangle — drag a corner or set the angle and size. The ratios stay fixed as the triangle grows, the special triangles give exact values, and the identities and complementary angles are checked with your numbers. Class 10 trigonometric ratios.

Heights & distances — a tower, a cliff and boat, a kite and a two-point survey. Drag the observer, the object or the top, and try the "match the angle" challenge. Class 10 applications of trigonometry.

Word problems — generated problems (easy, medium, hard) with a diagram, a hint, answer checking and a full worked solution. Class 10 heights and distances.

Part of Shunya. See the syllabus map for the chapters these modes map to. Also try the Function & Graph Explorer to plot any trigonometric function you like.

Formulas used in this lab

Trigonometric ratios of an acute angle Class 10

The three ratios: $\sin\theta = \dfrac{\text{opposite}}{\text{hypotenuse}},\quad \cos\theta = \dfrac{\text{adjacent}}{\text{hypotenuse}},\quad \tan\theta = \dfrac{\text{opposite}}{\text{adjacent}}$
Their reciprocals: $\operatorname{cosec}\theta = \dfrac{1}{\sin\theta},\quad \sec\theta = \dfrac{1}{\cos\theta},\quad \cot\theta = \dfrac{1}{\tan\theta}$
Tan as a quotient: $\tan\theta = \dfrac{\sin\theta}{\cos\theta},\qquad \cot\theta = \dfrac{\cos\theta}{\sin\theta}$
Pythagoras theorem: $\text{opposite}^2 + \text{adjacent}^2 = \text{hypotenuse}^2$
Sides from the hypotenuse h: $\text{opposite} = h\sin\theta,\qquad \text{adjacent} = h\cos\theta$
Complementary angles: $\sin(90^\circ - \theta) = \cos\theta,\quad \cos(90^\circ - \theta) = \sin\theta,\quad \tan(90^\circ - \theta) = \cot\theta$
Values at 0°, 30°, 45°, 60°, 90°: $\sin:\ 0,\ \tfrac12,\ \tfrac{1}{\sqrt2},\ \tfrac{\sqrt3}{2},\ 1 \qquad \cos:\ 1,\ \tfrac{\sqrt3}{2},\ \tfrac{1}{\sqrt2},\ \tfrac12,\ 0 \qquad \tan:\ 0,\ \tfrac{1}{\sqrt3},\ 1,\ \sqrt3,\ \text{not defined}$

Trigonometric identities Class 10

Pythagorean identity: $\sin^2\theta + \cos^2\theta = 1$
Tan and sec: $1 + \tan^2\theta = \sec^2\theta$
Cot and cosec: $1 + \cot^2\theta = \operatorname{cosec}^2\theta$
Tan as a quotient: $\tan\theta = \dfrac{\sin\theta}{\cos\theta}$
Complementary angles: $\sin(90^\circ - \theta) = \cos\theta,\qquad \tan(90^\circ - \theta) = \cot\theta$

Heights and distances Class 10

Angle from the sides: $\tan\theta = \dfrac{H}{d}\ \Rightarrow\ \theta = \tan^{-1}\dfrac{H}{d}$
Height from angle and distance (e = eye height, 0 at ground level): $H = d\tan\theta + e$
Distance from height and angle: $d = \dfrac{H}{\tan\theta}$
When a string or a ladder is involved: $\sin\theta = \dfrac{\text{height}}{\text{length}},\quad \cos\theta = \dfrac{\text{distance}}{\text{length}}$
Length from the height and angle: $\text{length} = \dfrac{\text{height}}{\sin\theta}$
Check with Pythagoras: $\text{length} = \sqrt{d^2 + H^2}$
Angle of depression = angle of elevation (alternate angles): $\theta_{\text{depression}} = \theta_{\text{elevation}}$
Two observation points, distance s apart: $H = \dfrac{s\,\tan\theta_1\tan\theta_2}{\tan\theta_2 - \tan\theta_1}$
Tower above a building: $\text{tower} = H_{\text{building}} + d\tan(\text{elevation})$
Handy values for 30°, 45°, 60°: $\tan 30^\circ = \tfrac{1}{\sqrt3},\quad \tan 45^\circ = 1,\quad \tan 60^\circ = \sqrt3$

The unit circle, radians and signs Class 11

A point on the unit circle: $P = (\cos\theta,\ \sin\theta),\qquad x^2 + y^2 = 1$
Radians and degrees: $\pi\ \text{rad} = 180^\circ,\qquad \theta_{\text{rad}} = \theta_{\text{deg}} \times \dfrac{\pi}{180}$
Tan from the coordinates: $\tan\theta = \dfrac{\sin\theta}{\cos\theta} = \dfrac{y}{x}$
Reference angle, by quadrant: $\text{I: } \theta \qquad \text{II: } 180^\circ - \theta \qquad \text{III: } \theta - 180^\circ \qquad \text{IV: } 360^\circ - \theta$
Positive ratios in quadrants I and II: $\text{I: all},\qquad \text{II: } \sin$
Positive ratios in quadrants III and IV: $\text{III: } \tan,\qquad \text{IV: } \cos$
Size from the reference angle: $|\sin\theta| = \sin(\text{ref. angle}),\quad |\cos\theta| = \cos(\text{ref. angle})$
Repeating every turn: $\sin(\theta + 360^\circ) = \sin\theta,\qquad \tan(\theta + 180^\circ) = \tan\theta$

Graphs of sin, cos and tan Class 11

The general transformed wave: $y = A\,f\bigl(B(x - C)\bigr) + D$
Amplitude: $|A|$
Period of sin and cos: $\dfrac{2\pi}{|B|}\quad\left(\dfrac{360^\circ}{|B|}\right)$
Period of tan: $\dfrac{\pi}{|B|}\quad\left(\dfrac{180^\circ}{|B|}\right)$
Phase shift and midline: $\text{shift right by } C,\qquad \text{midline } y = D$
Range of sin and cos: $-1 \le \sin\theta \le 1,\qquad -1 \le \cos\theta \le 1$
Where tan has asymptotes: $\cos\theta = 0\ \Rightarrow\ \theta = 90^\circ + 180^\circ n$
Cos is sin shifted: $\cos\theta = \sin(\theta + 90^\circ)$

Compound angles and trigonometric equations Class 11

Compound angles: $\sin(A\pm B) = \sin A\cos B \pm \cos A\sin B,\quad \cos(A\pm B) = \cos A\cos B \mp \sin A\sin B$
Tangent of a sum: $\tan(A\pm B) = \dfrac{\tan A \pm \tan B}{1 \mp \tan A\tan B}$
Double angle: $\sin 2A = 2\sin A\cos A,\quad \cos 2A = \cos^2 A - \sin^2 A = 2\cos^2 A - 1 = 1 - 2\sin^2 A$
Triple angle: $\sin 3A = 3\sin A - 4\sin^3 A,\quad \cos 3A = 4\cos^3 A - 3\cos A$
Sum to product: $\sin A + \sin B = 2\sin\tfrac{A+B}{2}\cos\tfrac{A-B}{2},\quad \cos A - \cos B = -2\sin\tfrac{A+B}{2}\sin\tfrac{A-B}{2}$
Harmonic form: $a\sin x + b\cos x = R\sin(x+\varphi),\ R=\sqrt{a^2+b^2},\ \tan\varphi = \tfrac ba$
General solutions: $\sin x=\sin\alpha \Rightarrow x=n\pi+(-1)^n\alpha;\ \cos x=\cos\alpha \Rightarrow x=2n\pi\pm\alpha;\ \tan x=\tan\alpha \Rightarrow x=n\pi+\alpha$