Trigonometry (III)

Syllabus tag: KCSE | Mathematics | Form 4 | Topic 4 Trigonometry (III)

Lesson objectives

By the end of this topic, you should be able to:

• Recall and define the trigonometric ratios.
• Derive and use the identity sin²x + cos²x = 1.
• Draw graphs of trigonometric functions.
• Solve trigonometric equations graphically and analytically.

Trigonometry (III)

This topic treats the trigonometric ratios as functions, draws their graphs, and solves equations involving them.

a) The ratios recalled

For a right-angled triangle: sin = opp/hyp, cos = adj/hyp, tan = opp/adj.

On the unit circle these extend to any angle. Here cos θ is the x-coordinate and sin θ the y-coordinate.

Note that tan θ = sin θ ÷ cos θ, which follows directly from the coordinates.

b) The Pythagorean identity

Since the unit circle has radius 1, any point on it satisfies x² + y² = 1.

Substituting gives sin²θ + cos²θ = 1.

Rearranged, sin²θ = 1 − cos²θ and cos²θ = 1 − sin²θ.

This converts between sines and cosines, which is how most trigonometric equations are reduced to a single ratio.

Note the notation: sin²θ means (sin θ)², not sin(θ²).

c) The graphs

y = sin x starts at 0 and rises to 1 at 90°. It returns to 0 at 180°, falls to −1 at 270°, then back to 0 at 360°.

y = cos x is the same shape shifted 90° left. It starts at 1.

y = tan x rises without limit, is undefined at 90° and 270°, and repeats every 180°.

The tangent graph has asymptotes where cos x is zero, since dividing by zero is undefined.

d) Amplitude and period

For y = a sin bx, the amplitude is |a| and the period is 360° ÷ b.

So y = 3 sin 2x oscillates between −3 and 3, completing a full cycle every 180°.

The amplitude stretches the graph vertically; b compresses it horizontally.

e) Solving equations graphically

Draw the trigonometric graph over the required range.

Draw the horizontal line for the given value.

The solutions are the x-coordinates of the intersections.

A question asking for solutions between 0° and 360° usually has two. Giving only one loses half the marks.

f) Solving analytically

Find the acute angle first, using tables or a calculator.

Then use the quadrant rules to find all the angles in the required range.

Take sin x = 0.5 between 0° and 360°. The acute angle is 30°. Sine is positive in the first and second quadrants, so x = 30° or 150°.

g) Equations needing the identity

If both sin and cos appear squared, use the identity to eliminate one.

Take 2cos²x + sin x = 2. Replace cos²x with 1 − sin²x. That gives 2 − 2sin²x + sin x = 2, so sin x(1 − 2sin x) = 0.

That gives sin x = 0 or sin x = 0.5, each with its own solutions.

Always check every solution lies within the range asked for.

h) Where this is used

Waves, tides and alternating current. Sound and light. Anything oscillating or repeating at a fixed period.

Words to know

• Period -- the interval after which a graph repeats.
• Amplitude -- the maximum distance of a wave from its centre line.
• Asymptote -- a line a curve approaches but never reaches.
• Identity -- an equation true for all values of the variable.
• Related acute angle -- the acute angle used with CAST to find all solutions.

:::checkpoint Check yourself

1. State the Pythagorean identity.
2. What is the period of y = tan x?
3. Find the amplitude and period of y = 4 sin 3x.
4. Solve sin x = 0.5 for 0° ≤ x ≤ 360°. :::

Bridge to practice

The exercises begin with the three graphs and their features, move through amplitude, period and the identity, and finish with solving equations including quadratic forms. For every equation, write down the range first and check how many solutions it should contain.