Trigonometric Functions

The One Thing

Trigonometric functions relate angles to right-triangle side ratios and to coordinates on the unit circle. Their evaluation and manipulation depend on angle measure, quadrant signs, periodicity, identities, and compound-angle formulas, with all domain restrictions observed.

Definitions and Results

• Angle: A figure formed by rotating a ray about its endpoint. Anticlockwise rotation gives positive angles; clockwise rotation gives negative angles.
• Degree measure: An angle-measurement system in which one complete revolution is $360^{\circ }$.
• Radian measure: The angle subtended at the centre of a circle by an arc whose length equals the radius. One complete revolution is $2\pi$ radians.
• Conversion of angle measure:

$$180^{\circ }=\pi \text{ radians},{\theta }^{\circ }=\frac{\theta \pi }{180}\text{ radians}.$$ Hence $1^{\circ }=\pi /180$ radians and $1$ radian $=180^{\circ }/\pi$.

• Trigonometric functions: Functions such as sine, cosine, tangent, cosecant, secant, and cotangent that associate an angle or real number with a value.
• Right-triangle definitions: For an acute angle $\theta$,

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

• Unit circle: A circle of radius $1$ centred at the origin. The point corresponding to an angle $\theta$ has coordinates

$$(\cos \theta ,\sin \theta ).$$

• Sine function: $\sin \theta$ is the $y$-coordinate of the corresponding unit-circle point. Its domain is $\mathbb{R}$, range is $[-1,1]$, and fundamental period is $2\pi$.
• Cosine function: $\cos \theta$ is the $x$-coordinate of the corresponding unit-circle point. Its domain is $\mathbb{R}$, range is $[-1,1]$, and fundamental period is $2\pi$.
• Tangent function:

$$\tan \theta =\frac{\sin \theta }{\cos \theta },$$ defined only when $\cos \theta \ne 0$. Its domain is $$\mathbb{R}\setminus \left\{\frac{(2n+1)\pi }{2}:n\in \mathbb{Z}\right\},$$ range is $\mathbb{R}$, and fundamental period is $\pi$.

• Reciprocal functions:

$$\mathrm{cosec}\theta =\frac{1}{\sin \theta },\sec \theta =\frac{1}{\cos \theta },\cot \theta =\frac{1}{\tan \theta }=\frac{\cos \theta }{\sin \theta },$$ wherever the expressions are defined.

• Domains and ranges:

$$\begin{array}{ccc}\text{Function}& \text{Domain}& \text{Range}\\ \sin x,\cos x& \mathbb{R}& [-1,1]\\ \tan x& \mathbb{R}\setminus \{(2n+1)\pi /2:n\in \mathbb{Z}\}& \mathbb{R}\\ \sec x& \mathbb{R}\setminus \{(2n+1)\pi /2:n\in \mathbb{Z}\}& (-\infty ,-1]\cup [1,\infty )\\ \mathrm{cosec}x& \mathbb{R}\setminus \{n\pi :n\in \mathbb{Z}\}& (-\infty ,-1]\cup [1,\infty )\\ \cot x& \mathbb{R}\setminus \{n\pi :n\in \mathbb{Z}\}& \mathbb{R}\end{array}$$

• Trigonometric identities: Equations involving trigonometric functions that remain true for every angle for which both sides are defined.
• Fundamental identities:

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

• Complementary-angle relations:

$$\sin \left(\frac{\pi }{2}-\theta \right)=\cos \theta ,\cos \left(\frac{\pi }{2}-\theta \right)=\sin \theta ,$$ $$\tan \left(\frac{\pi }{2}-\theta \right)=\cot \theta ,\sec \left(\frac{\pi }{2}-\theta \right)=\mathrm{cosec}\theta .$$

• Signs by quadrant: Sine is positive in quadrants I and II; cosine is positive in quadrants I and IV; tangent is positive in quadrants I and III. Reciprocal functions have the same signs as their corresponding functions.
• Even and odd functions: Cosine and secant are even:

$$f(-\theta )=f(\theta ).$$ Sine, cosecant, tangent, and cotangent are odd: $$f(-\theta )=-f(\theta ).$$ In particular, $$\sin (-\theta )=-\sin \theta ,\cos (-\theta )=\cos \theta ,\tan (-\theta )=-\tan \theta ,$$ $$\mathrm{cosec}(-\theta )=-\mathrm{cosec}\theta ,\sec (-\theta )=\sec \theta ,\cot (-\theta )=-\cot \theta .$$

• Periodicity: A function is periodic if its values repeat after a fixed positive interval. The least positive period is its fundamental period:

$$\sin (\theta +2\pi )=\sin \theta ,\cos (\theta +2\pi )=\cos \theta ,$$ $$\tan (\theta +\pi )=\tan \theta .$$ The fundamental periods of sine, cosine, and tangent are $2\pi$, $2\pi$, and $\pi$, respectively.

• Compound angle: An angle written as the sum or difference of two angles, such as $A+B$ or $A-B$.
• Compound-angle formulas:

$$\sin (A+B)=\sin A\cos B+\cos A\sin B,$$ $$\sin (A-B)=\sin A\cos B-\cos A\sin B,$$ $$\cos (A+B)=\cos A\cos B-\sin A\sin B,$$ $$\cos (A-B)=\cos A\cos B+\sin A\sin B,$$ $$\tan (A+B)=\frac{\tan A+\tan B}{1-\tan A\tan B},$$ $$\tan (A-B)=\frac{\tan A-\tan B}{1+\tan A\tan B},$$ whenever the expressions are defined.

• Double-angle formulas:

$$\sin 2A=2\sin A\cos A,$$ $$\cos 2A={\cos }^{2}A-{\sin }^{2}A=1-2{\sin }^{2}A=2{\cos }^{2}A-1,$$ $$\tan 2A=\frac{2\tan A}{1-{\tan }^{2}A},$$ where defined.

• Common exact values:

$$\sin 0=0,\cos 0=1,\tan 0=0;$$ $$\sin \frac{\pi }{6}=\frac{1}{2},\cos \frac{\pi }{6}=\frac{\sqrt{3}}{2},\tan \frac{\pi }{6}=\frac{1}{\sqrt{3}};$$ $$\sin \frac{\pi }{4}=\cos \frac{\pi }{4}=\frac{1}{\sqrt{2}},\tan \frac{\pi }{4}=1;$$ $$\sin \frac{\pi }{3}=\frac{\sqrt{3}}{2},\cos \frac{\pi }{3}=\frac{1}{2},\tan \frac{\pi }{3}=\sqrt{3};$$ $$\sin \frac{\pi }{2}=1,\cos \frac{\pi }{2}=0.$$

• General solutions: For any integer $n$,

$$\sin x=0⟹x=n\pi ,$$ $$\cos x=0⟹x=\frac{(2n+1)\pi }{2},$$ $$\tan x=0⟹x=n\pi .$$

Worked Methods

Converting between degrees and radians

• Use

$$180^{\circ }=\pi \text{ radians}.$$

• To convert ${\theta }^{\circ }$ to radians, multiply by $\pi /180$:

$${\theta }^{\circ }=\frac{\theta \pi }{180}.$$

• To convert radians to degrees, multiply by $180/\pi$:

$$\theta \text{ radians}={\frac{180\theta }{\pi }}^{\circ }.$$

Evaluating trigonometric functions using a right-angled triangle

• Identify the angle $\theta$.
• Identify the perpendicular, base, and hypotenuse relative to $\theta$.
• Apply

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

Using the unit circle

• Represent the angle $\theta$ as a rotation from the positive $x$-axis.
• Locate the corresponding point on the unit circle.
• Read its coordinates as

$$(\cos \theta ,\sin \theta ).$$

• Determine signs from the quadrant.
• Use the coordinate values to obtain sine and cosine; then use

$$\tan \theta =\frac{\sin \theta }{\cos \theta }$$ when $\cos \theta \ne 0$.

• The unit circle explains the functions’ signs, periodicity, exact values, and graphs for all real angles.

Simplifying expressions with identities

• Rewrite reciprocal functions using

$$\mathrm{cosec}\theta =\frac{1}{\sin \theta },\sec \theta =\frac{1}{\cos \theta },\cot \theta =\frac{\cos \theta }{\sin \theta }.$$

• Rewrite tangent or cotangent using

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

• Apply a fundamental identity:

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

• Check that no denominator is zero.
• Distinguish an identity, which is true for every permitted angle, from an equation true only for particular angle values.

Determining signs and exact values

• Reduce the angle to a reference angle.
• Identify its quadrant.
• Use the quadrant sign rules:

- sine positive in quadrants I and II; - cosine positive in quadrants I and IV; - tangent positive in quadrants I and III.

• Use the corresponding exact value, such as

$$\sin \frac{\pi }{6}=\frac{1}{2},\cos \frac{\pi }{3}=\frac{1}{2},\tan \frac{\pi }{4}=1.$$

• Apply the same sign to reciprocal functions.

Applying compound-angle formulas

• Express the angle as $A+B$ or $A-B$.
• Substitute into the appropriate formula:

$$\sin (A+B)=\sin A\cos B+\cos A\sin B,$$ $$\sin (A-B)=\sin A\cos B-\cos A\sin B,$$ $$\cos (A+B)=\cos A\cos B-\sin A\sin B,$$ $$\cos (A-B)=\cos A\cos B+\sin A\sin B.$$

• For tangent, use

$$\tan (A+B)=\frac{\tan A+\tan B}{1-\tan A\tan B},$$ \[ \tan(A-B)=\frac{\tan A-\tan B}{1+\tan A\tan B}.