Trigonometry lessons

Inside a right-angled triangle

A right-angled triangle has one angle of exactly 90°. The side opposite this right angle is the longest side, called the hypotenuse.

The other two sides are named relative to the angle you are working with. The side across from the angle is the opposite. The side next to the angle (that is not the hypotenuse) is the adjacent.

Interactive triangle side explorer.

Practice: Label the sides

Label the sides of a right-angled triangle.

SOH CAH TOA

The three trig ratios relate an angle to two sides of a right-angled triangle. The mnemonic SOH CAH TOA helps you remember them.

1. \sin(\theta) = \frac{\textcolor{#8b5cf6}{opposite}}{\textcolor{#f871b9}{hypotenuse}} (SOH)
2. \cos(\theta) = \frac{\textcolor{#3b82f6}{adjacent}}{\textcolor{#f871b9}{hypotenuse}} (CAH)
3. \tan(\theta) = \frac{\textcolor{#8b5cf6}{opposite}}{\textcolor{#3b82f6}{adjacent}} (TOA)

Each ratio is a fraction. If you know the angle and one side, you can find the other side. If you know two sides, you can find the angle.

Choosing the right ratio

Look at which two sides are involved in your problem. If you have the opposite and hypotenuse, use sine. If you have adjacent and hypotenuse, use cosine. If you have opposite and adjacent, use tangent.

Interactive ratio explorer.

Practice: Pick the ratio

Pick the correct trig ratio.

Finding a missing side

Once you pick the right ratio, substitute the known values and solve for the unknown side. The key step is rearranging the fraction.

1. \sin(35°) = \frac{\textcolor{#8b5cf6}{x}}{\textcolor{#f871b9}{10}} (set up the ratio)
2. \textcolor{#8b5cf6}{x} = \textcolor{#f871b9}{10} \times \sin(35°) (multiply both sides by 10)
3. \textcolor{#8b5cf6}{x} = 5.74 (evaluate with a calculator)

Always check: the opposite side must be shorter than the hypotenuse. If your answer is larger, something went wrong.

The ratio slider

See how changing the angle affects the trig ratios and the triangle shape in real time.

Ratio slider — live trig ratios as angle changes.

Practice: Find the missing side

Find the missing side practice.

Finding a missing angle

When you know two sides but not the angle, use the inverse trig functions. These undo sine, cosine, or tangent to give you the angle back.

Inverse sine: \theta = \sin^{-1}\left(\frac{opposite}{hypotenuse}\right)

The notation \sin^{-1} is also written as \arcsin. Your calculator may use either label. They mean the same thing.

Working backwards

Set up the ratio with two known sides, then apply the inverse function to both sides.

1. \tan(\textcolor{#f871b9}{\theta}) = \frac{7}{4} (opposite = 7, adjacent = 4)
2. \textcolor{#f871b9}{\theta} = \tan^{-1}\left(\frac{7}{4}\right) (apply inverse tan)
3. \textcolor{#f871b9}{\theta} = 60.26° (evaluate)

Inverse angle worked example.

Practice: Find the missing angle

Find the missing angle practice.

Degrees and radians

Degrees and radians are two ways to measure angles. A full circle is 360° or 2\pi radians.

Key conversion: \pi\ radians = 180°

To convert degrees to radians, multiply by \frac{\pi}{180}. To go the other way, multiply by \frac{180}{\pi}.

The angle converter

Common angles have clean radian values. Try converting a few to build intuition.

1. 90° = 90 \times \frac{\pi}{180} = \frac{\pi}{2} (quarter turn)
2. 60° = 60 \times \frac{\pi}{180} = \frac{\pi}{3} (sixth of a circle)
3. 45° = 45 \times \frac{\pi}{180} = \frac{\pi}{4} (eighth of a circle)

Draggable angle converter.

Practice: Convert angles

Convert angles practice.

The unit circle

The unit circle is a circle with radius 1 centred at the origin. It extends trig beyond triangles to any angle, including negative angles and angles greater than 360°.

For any angle \theta, the point on the circle has coordinates (\cos\theta,\ \sin\theta). The x-coordinate is cosine and the y-coordinate is sine.

Unit circle explorer.

Quadrants and signs

The unit circle is divided into four quadrants. In quadrant I (top-right), both sine and cosine are positive. In quadrant II, sine is positive but cosine is negative.

In quadrant III, both are negative. In quadrant IV, cosine is positive and sine is negative. A common mnemonic is All Students Take Calculus for which ratios are positive in each quadrant.

Practice: Read the unit circle

Read the unit circle practice.

Trig graphs

Plotting \sin(\theta) and \cos(\theta) against \theta produces smooth, repeating wave shapes. These waves repeat every 2\pi radians (or 360°).

The amplitude is the height from the centre line to a peak. The period is the distance along the x-axis for one full cycle.

Wave builder — sliders for A, B, C, D.

Shaping the wave

The general form y = A\sin(Bx + C) + D controls every aspect of the wave. Each letter changes one property.

1. \textcolor{#f871b9}{A} = amplitude (height of the wave)
2. \textcolor{#8b5cf6}{B} \to period = \frac{2\pi}{\textcolor{#8b5cf6}{B}} (horizontal stretch)
3. \textcolor{#3b82f6}{C} \to phase\ shift = -\frac{\textcolor{#3b82f6}{C}}{B} (left/right slide)
4. \textcolor{#22c55e}{D} = vertical\ shift (up/down shift)

Practice: Match the wave

Match the wave practice.

The Pythagorean identity

The most important trig identity comes directly from the unit circle and the Pythagorean theorem.

Pythagorean identity: \sin^2\theta + \cos^2\theta = 1

Since the point (\cos\theta,\ \sin\theta) sits on a circle of radius 1, the sum of the squares of its coordinates must equal 1. This identity is true for every angle.

Using the identity

If you know sine, you can find cosine (and vice versa) without knowing the angle. Just rearrange the identity.

1. \sin\theta = \frac{3}{5} (given)
2. \left(\frac{3}{5}\right)^2 + \cos^2\theta = 1 (substitute into identity)
3. \cos^2\theta = 1 - \frac{9}{25} = \frac{16}{25} (rearrange)
4. \cos\theta = \pm\frac{4}{5} (take square root)

The \pm sign depends on which quadrant the angle is in. In the first quadrant, both sine and cosine are positive.

Pythagorean identity explorer.

Practice: Apply the identity

Apply the Pythagorean identity practice.

Beyond right-angled triangles

The sine rule works on any triangle, not just right-angled ones. It connects each side to the sine of its opposite angle.

Sine rule: \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}

Sine rule worked example.

Applying the sine rule

Use the sine rule when you know a side and its opposite angle, plus one other measurement. Pick the two fractions that contain your known and unknown values, then solve.

1. \frac{\textcolor{#f871b9}{x}}{\sin 40°} = \frac{8}{\sin 75°} (set up two fractions)
2. \textcolor{#f871b9}{x} = \frac{8 \times \sin 40°}{\sin 75°} (multiply both sides by sin 40)
3. \textcolor{#f871b9}{x} = 5.32 (evaluate)

Practice: Use the sine rule

Use the sine rule practice.

The cosine rule

When you know two sides and the angle between them, the sine rule cannot help. The cosine rule handles this case.

Cosine rule: c^2 = a^2 + b^2 - 2ab\cos C

This is a generalisation of the Pythagorean theorem. When C = 90°, the \cos C term vanishes and you get c^2 = a^2 + b^2.

Cosine rule worked example.

Cosine rule worked example

Find side c given a = 6, b = 8, and C = 50°.

1. c^2 = \textcolor{#f871b9}{6}^2 + \textcolor{#8b5cf6}{8}^2 - 2(\textcolor{#f871b9}{6})(\textcolor{#8b5cf6}{8})\cos 50° (substitute)
2. c^2 = 36 + 64 - 96 \times 0.6428 (evaluate squares)
3. c^2 = 100 - 61.71 = 38.29 (simplify)
4. c = \sqrt{38.29} = 6.19 (take square root)

Practice: Use the cosine rule

Use the cosine rule practice.

Area of any triangle

You can find the area of any triangle without knowing the height. Use two sides and the sine of the included angle.

Triangle area formula: Area = \frac{1}{2}ab\sin C

Triangle area explorer.

Why the formula works

The formula works because b\sin C gives the perpendicular height of the triangle when a is the base. Multiplying base by height and halving gives the area.

Notice that the area is zero when C = 0° or C = 180° (the triangle collapses to a line) and maximum when C = 90°.

Practice: Calculate the area

Calculate the area practice.

Trig identities overview

Beyond the Pythagorean identity, several other relationships are always true. These let you rewrite trig expressions in simpler forms.

1. \tan\theta = \frac{\sin\theta}{\cos\theta} (tangent ratio)
2. \sin^2\theta + \cos^2\theta = 1 (Pythagorean identity)
3. 1 + \tan^2\theta = \sec^2\theta (divide by cos^2)
4. 1 + \cot^2\theta = \csc^2\theta (divide by sin^2)

These identities are tools for simplifying complex expressions and solving trig equations.

Practice: Simplify the expression

Simplify expression practice.