Trigonometry

The One Thing

Trigonometry connects angles and lengths through ratios defined in right-angled triangles. Identities transform expressions into equivalent forms, while heights and distances problems apply sine, cosine, tangent and the Pythagorean theorem to labelled geometric models.

Definitions and Results

• Trigonometric ratio: A relationship between two sides of a right-angled triangle for a given acute angle.
• Perpendicular: The side opposite the chosen angle in a right-angled triangle.
• Base or adjacent side: The side next to the chosen angle, excluding the hypotenuse.
• Hypotenuse: The side opposite the right angle; it is the longest side of the triangle.
• Sine: For an acute angle $\theta$, $\sin (\theta )=\frac{\text{perpendicular}}{\text{hypotenuse}}$.
• Cosine: For an acute angle $\theta$, $\cos (\theta )=\frac{\text{base}}{\text{hypotenuse}}$.
• Tangent: For an acute angle $\theta$, $\tan (\theta )=\frac{\text{perpendicular}}{\text{base}}$.
• Cosecant: The reciprocal of sine: $\mathrm{cosec}(\theta )=\frac{\text{hypotenuse}}{\text{perpendicular}}=\frac{1}{\sin (\theta )}$.
• Secant: The reciprocal of cosine: $\sec (\theta )=\frac{\text{hypotenuse}}{\text{base}}=\frac{1}{\cos (\theta )}$.
• Cotangent: The reciprocal of tangent: $\cot (\theta )=\frac{\text{base}}{\text{perpendicular}}=\frac{1}{\tan (\theta )}$.
• Trigonometric identity: An equation involving trigonometric ratios that is true for every angle for which both sides are defined.
• Fundamental identities:

$${\sin }^2(\theta )+{\cos }^2(\theta )=1,$$ $$1+{\tan }^2(\theta )={\sec }^2(\theta ),$$ $$1+{\cot }^2(\theta )={\mathrm{cosec}}^2(\theta ).$$

• Derived identities:

$$1-{\sin }^2(\theta )={\cos }^2(\theta ),$$ $$1-{\cos }^2(\theta )={\sin }^2(\theta ),$$ $${\sec }^2(\theta )-1={\tan }^2(\theta ),$$ $${\mathrm{cosec}}^2(\theta )-1={\cot }^2(\theta ).$$

• Quotient identities: Whenever the expressions are defined,

$$\tan (\theta )=\frac{\sin (\theta )}{\cos (\theta )},\cot (\theta )=\frac{\cos (\theta )}{\sin (\theta )}.$$

• Complementary-angle relations:

$$\sin (90^{\circ }-\theta )=\cos (\theta ),$$ $$\cos (90^{\circ }-\theta )=\sin (\theta ),$$ $$\tan (90^{\circ }-\theta )=\cot (\theta ),$$ $$\sec (90^{\circ }-\theta )=\mathrm{cosec}(\theta ),$$ $$\mathrm{cosec}(90^{\circ }-\theta )=\sec (\theta ).$$

• Standard sine values:

$$\sin (0^{\circ })=0,\sin (30^{\circ })=\frac{1}{2},\sin (45^{\circ })=\frac{1}{\sqrt{2}},$$ $$\sin (60^{\circ })=\frac{\sqrt{3}}{2},\sin (90^{\circ })=1.$$

• Standard cosine values:

$$\cos (0^{\circ })=1,\cos (30^{\circ })=\frac{\sqrt{3}}{2},\cos (45^{\circ })=\frac{1}{\sqrt{2}},$$ $$\cos (60^{\circ })=\frac{1}{2},\cos (90^{\circ })=0.$$

• Standard tangent values:

$$\tan (0^{\circ })=0,\tan (30^{\circ })=\frac{1}{\sqrt{3}},\tan (45^{\circ })=1,$$ $$\tan (60^{\circ })=\sqrt{3},$$ while $\tan (90^{\circ })$ is undefined.

• Reciprocal standard values: Values of $\mathrm{cosec}$, $\sec$ and $\cot$ are obtained by taking the reciprocals of the corresponding sine, cosine and tangent values, where defined.
• Angle of elevation: The angle between the horizontal line and the line of sight when an object is viewed above the observer’s eye level.
• Angle of depression: The angle between the horizontal line and the line of sight when an object is viewed below the observer’s eye level.
• Line of sight: The straight line joining the observer’s eye to the object being observed.
• Horizontal distance: The distance measured along a level or horizontal line between the observer and the foot of the object.
• Pythagorean theorem:

$${\text{hypotenuse}}^2={\text{perpendicular}}^2+{\text{base}}^2.$$

• Conditions for identity use: An identity applies for all permitted angles, but denominators must be non-zero and both sides must be defined.
• Conditions for heights and distances: Angles of elevation and depression are measured from a horizontal line. The observer’s eye height must be included when the total height of an object above the ground is required.

Worked Methods

Proving a trigonometric identity

• Begin with the more complicated side where possible.
• Simplify using valid algebraic operations and the fundamental identities.
• Convert ratios into sine and cosine if this makes the expression easier to simplify.
• Continue until the expression becomes the other side.
• Ensure that all denominators remain non-zero and that the identity is used only where both sides are defined.

For example, expressions involving ${\sec }^2(\theta )-1$ can be changed using $$1+{\tan }^2(\theta )={\sec }^2(\theta ),$$ giving $${\sec }^2(\theta )-1={\tan }^2(\theta ).$$

An identity is not solved for one particular angle; it remains valid for every permitted value of $\theta$.

Selecting a ratio in a right-angled triangle

• Draw and label the right-angled triangle.
• Mark the chosen angle $\theta$.
• Identify the perpendicular, base or adjacent side, and hypotenuse relative to $\theta$.
• Select the ratio that contains the known and unknown sides:

$$\sin (\theta )=\frac{\text{perpendicular}}{\text{hypotenuse}},$$ $$\cos (\theta )=\frac{\text{base}}{\text{hypotenuse}},$$ $$\tan (\theta )=\frac{\text{perpendicular}}{\text{base}}.$$

• Substitute the measurements and rearrange to find the required quantity.
• Use degree mode when the angle is given in degrees, maintain consistent units, and round the final answer only as instructed.

Solving a heights and distances problem

• Translate the situation into a labelled right-angled triangle.
• Identify the horizontal line and the line of sight.
• Mark the angle of elevation or angle of depression from the horizontal line.
• Identify the perpendicular height, horizontal distance and, where relevant, the line of sight.
• Choose the appropriate ratio:

- If height and horizontal distance are involved, use $$\tan (\theta )=\frac{\text{height}}{\text{horizontal distance}}.$$ - If height and line of sight are involved, use $$\sin (\theta )=\frac{\text{height}}{\text{line of sight}}.$$ - If horizontal distance and line of sight are involved, use $$\cos (\theta )=\frac{\text{horizontal distance}}{\text{line of sight}}.$$

• Rearrange and calculate the required length or angle.
• If an angle of depression is given, use the horizontal lines and alternate interior angles to identify the corresponding angle of elevation when appropriate.
• Add the observer’s eye height when the question asks for the total height of the object above the ground.
• Check units and round only at the final stage.

Combining the Pythagorean theorem with trigonometry

• Identify the right-angled triangle and its three sides.
• Use

$${\text{hypotenuse}}^2={\text{perpendicular}}^2+{\text{base}}^2$$ when two sides are known and the third is required.

• Use the resulting side length with sine, cosine or tangent if an angle or another side must then be found.
• Confirm that the hypotenuse is opposite the right angle and is the longest side.

Where It Goes Wrong

• Treating a trigonometric identity as an equation to solve for a particular angle, rather than recognising that it is true for all permitted angles.
• Beginning an identity proof with invalid algebraic operations or cancelling a denominator without checking that it is non-zero.
• Misidentifying the perpendicular, base or hypotenuse after choosing the angle; these labels depend on the chosen angle.
• Measuring an angle of elevation or depression from the line of sight instead of from the horizontal line.
• Choosing a ratio that does not match the known and unknown sides, such as using sine when the relevant sides are the perpendicular and base.
• Omitting the observer’s eye height when the required quantity is the total height above the ground, or using incorrect units, radian mode, or premature rounding.

What Gets Asked

• State or use the definitions of sine, cosine, tangent, cosecant, secant and cotangent in a right-angled triangle.
• Evaluate standard trigonometric values for $0^{\circ }$, $30^{\circ }$, $45^{\circ }$, $60^{\circ }$ and $90^{\circ }$, including identifying that $\tan (90^{\circ })$ is undefined.
• Use reciprocal, quotient and complementary-angle identities.
• Prove or simplify a trigonometric identity using

$${\sin }^2(\theta )+{\cos }^2(\theta )=1,1+{\tan }^2(\theta )={\sec }^2(\theta ),1+{\cot }^2(\theta )={\mathrm{cosec}}^2(\theta ).$$

• Find an unknown side or angle in a right-angled triangle by selecting sine, cosine or tangent.
• Solve heights and distances problems involving an angle of elevation, an angle of depression, a line of sight and horizontal distance.
• Use alternate interior angles to relate an angle of depression to the corresponding angle of elevation.
• Include an observer’s eye height in a total-height calculation.
• Combine the Pythagorean theorem,

$${\text{hypotenuse}}^2={\text{perpendicular}}^2+{\text{base}}^2,$$ with trigonometric ratios.