Trigonometric Functions

Trigonometric functions describe relationships between angles and numerical values. They are fundamental to mathematics because they allow us to calculate unknown lengths, analyse repeating patterns, model circular motion, and solve problems involving direction and rotation.

The six trigonometric functions are:

• Sine (sin)

• Cosine (cos)

• Tangent (tan)

• Cosecant (csc)

• Secant (sec)

• Cotangent (cot)

These functions can be defined using right triangles or the unit circle. The unit circle extends trigonometry beyond acute angles, allowing us to evaluate functions for negative angles, angles greater than 360°, and angles measured in radians.

Trigonometric functions are widely used in physics, engineering, navigation, astronomy, computer graphics, and the study of waves and oscillations.

The Six Trigonometric Functions

In a right triangle, the sides are identified relative to a chosen acute angle θ.

• Opposite: The side directly across from θ.

• Adjacent: The side next to θ that is not the hypotenuse.

• Hypotenuse: The longest side, opposite the right angle.

The three primary trigonometric functions are:

sin θ = opposite/hypotenuse

cos θ = adjacent/hypotenuse

tan θ = opposite/adjacent

A common memory aid is SOH–CAH–TOA.

The remaining three functions are reciprocals of the primary functions.

csc θ = hypotenuse/opposite = 1/sin θ

sec θ = hypotenuse/adjacent = 1/cos θ

cot θ = adjacent/opposite = 1/tan θ

These triangle definitions apply directly to acute angles. The unit circle provides definitions for more general angles.

Trigonometric Functions on the Unit Circle

The unit circle has radius 1 and is centred at the origin.

Its equation is:

x² + y² = 1

For an angle θ in standard position, the point where its terminal side meets the unit circle has coordinates:

(x, y) = (cos θ, sin θ)

Therefore:

• cos θ = x

• sin θ = y

• tan θ = y/x

• csc θ = 1/y

• sec θ = 1/x

• cot θ = x/y

The last four expressions are defined only when their denominators are nonzero.

Example

Suppose a point on the unit circle is:

(3/5, 4/5)

Then:

cos θ = 3/5

sin θ = 4/5

tan θ = (4/5)/(3/5) = 4/3

csc θ = 5/4

sec θ = 5/3

cot θ = 3/4

All six trigonometric functions can be determined from a single point.

Sine Function

The sine function describes the vertical coordinate of a point on the unit circle.

sin θ = y

For an acute angle in a right triangle:

sin θ = opposite/hypotenuse

Important Properties

• Domain: All real numbers

• Range: −1 ≤ sin θ ≤ 1

• Period: 360° or 2π radians

• Maximum value: 1

• Minimum value: −1

Sine is positive in Quadrants I and II.

Sine is negative in Quadrants III and IV.

Common Values

The sine graph is a smooth repeating wave that begins at zero and initially increases.

Cosine Function

The cosine function describes the horizontal coordinate of a point on the unit circle.

cos θ = x

For an acute angle in a right triangle:

cos θ = adjacent/hypotenuse

Important Properties

• Domain: All real numbers

• Range: −1 ≤ cos θ ≤ 1

• Period: 360° or 2π radians

• Maximum value: 1

• Minimum value: −1

Cosine is positive in Quadrants I and IV.

Cosine is negative in Quadrants II and III.

Common Values

The cosine graph begins at its maximum value of 1.

Like sine, cosine repeats every complete revolution.

Tangent Function

The tangent function is the ratio of sine to cosine.

tan θ = sin θ/cos θ

On the unit circle:

tan θ = y/x

For an acute angle in a right triangle:

tan θ = opposite/adjacent

Important Properties

• Domain: All real angles except 90° + 180°n, where n is any integer

• Range: All real numbers

• Period: 180° or π radians

• Zeros: 180°n, where n is any integer

Tangent is positive in Quadrants I and III.

Tangent is negative in Quadrants II and IV.

Why Tangent Can Be Undefined

At 90°:

cos 90° = 0

Therefore:

tan 90° = sin 90°/cos 90° = 1/0

Division by zero is undefined.

The same occurs at 270° and coterminal angles.

The tangent graph has vertical asymptotes where cosine is zero.

Unlike sine and cosine, tangent is not restricted to values between −1 and 1.

Cosecant Function

Cosecant is the reciprocal of sine.

csc θ = 1/sin θ

On the unit circle:

csc θ = 1/y

Important Properties

• Domain: All real angles except 180°n

• Range: csc θ ≤ −1 or csc θ ≥ 1

• Period: 360° or 2π radians

Cosecant is undefined whenever sine equals zero.

For example:

csc 0° is undefined.

csc 180° is undefined.

Worked Example

Find csc 30°.

Since:

sin 30° = 1/2

Then:

csc 30° = 1/(1/2)

csc 30° = 2

The cosecant graph consists of separate curved branches because the function is undefined wherever sine is zero.

Secant Function

Secant is the reciprocal of cosine.

sec θ = 1/cos θ

On the unit circle:

sec θ = 1/x

Important Properties

• Domain: All real angles except 90° + 180°n

• Range: sec θ ≤ −1 or sec θ ≥ 1

• Period: 360° or 2π radians

Secant is undefined whenever cosine equals zero.

Worked Example

Find sec 60°.

Since:

cos 60° = 1/2

Then:

sec 60° = 1/(1/2)

sec 60° = 2

The secant graph has separate branches and vertical asymptotes at angles where cosine is zero.

Cotangent Function

Cotangent is the reciprocal of tangent.

cot θ = cos θ/sin θ

On the unit circle:

cot θ = x/y

For an acute angle in a right triangle:

cot θ = adjacent/opposite

Important Properties

• Domain: All real angles except 180°n

• Range: All real numbers

• Period: 180° or π radians

• Zeros: 90° + 180°n

Cotangent is undefined when sine equals zero.

Worked Example

Find cot 45°.

Since:

tan 45° = 1

Then:

cot 45° = 1/1

cot 45° = 1

The cotangent graph has decreasing branches between its vertical asymptotes.

Comparing the Six Trigonometric Functions

The following table summarises their definitions, domains, ranges, and periods.

Here, n represents any integer.

In radians:

• 180°n becomes nπ.

• 90° + 180°n becomes π/2 + nπ.

• 360° becomes 2π.

• 180° becomes π.

Understanding these restrictions is essential when evaluating trigonometric expressions.

Exact Values for Common Angles

The most important reference angles are:

0°, 30°, 45°, 60°, and 90°.

Their exact sine, cosine, and tangent values are:

The remaining functions can be found by taking reciprocals.

For example:

sin 60° = √3/2

Therefore:

csc 60° = 2/√3 = 2√3/3

Signs of Trigonometric Functions in Each Quadrant

The signs of trigonometric functions depend on the signs of the x- and y-coordinates.

Quadrant I

x > 0 and y > 0

All six trigonometric functions are positive.

Quadrant II

x < 0 and y > 0

Sine and cosecant are positive.

Cosine, secant, tangent, and cotangent are negative.

Quadrant III

x < 0 and y < 0

Tangent and cotangent are positive.

Sine, cosecant, cosine, and secant are negative.

Quadrant IV

x > 0 and y < 0

Cosine and secant are positive.

Sine, cosecant, tangent, and cotangent are negative.

Complete Sign Table

The reciprocal of a nonzero number has the same sign as the original number.

Therefore, sine and cosecant always share the same sign wherever defined, as do cosine and secant, and tangent and cotangent.

Reference Angles

A reference angle is the acute angle formed between an angle's terminal side and the x-axis.

Reference angles allow us to determine trigonometric values for angles outside the first quadrant.

For 0° ≤ θ < 360°:

• Quadrant I: α = θ

• Quadrant II: α = 180° − θ

• Quadrant III: α = θ − 180°

• Quadrant IV: α = 360° − θ

Quadrantal angles are evaluated directly from the unit circle.

Worked Example: sin 150°

150° lies in Quadrant II.

Reference angle:

180° − 150° = 30°

Sine is positive in Quadrant II.

Therefore:

sin 150° = 1/2

Worked Example: cos 240°

240° lies in Quadrant III.

Reference angle:

240° − 180° = 60°

Cosine is negative in Quadrant III.

Therefore:

cos 240° = −1/2

Worked Example: tan 315°

315° lies in Quadrant IV.

Reference angle:

360° − 315° = 45°

Tangent is negative in Quadrant IV.

Therefore:

tan 315° = −1

Evaluating Trigonometric Functions for Any Angle

Trigonometric functions can be evaluated for negative angles and angles greater than 360°.

Coterminal Angles

Angles that differ by a whole number of complete revolutions share the same terminal side.

For example:

420° − 360° = 60°

Therefore:

sin 420° = sin 60°

sin 420° = √3/2

Negative Angles

Negative angles are measured clockwise.

For example:

−60° is coterminal with 300°.

Therefore:

cos(−60°) = cos 300°

cos(−60°) = 1/2

Worked Example: sec(−120°)

First find a positive coterminal angle:

−120° + 360° = 240°

The reference angle is:

240° − 180° = 60°

Cosine is negative in Quadrant III.

cos 240° = −1/2

Therefore:

sec(−120°) = 1/(−1/2)

sec(−120°) = −2

Evaluating Functions in Radians

Trigonometric functions are often evaluated using radians rather than degrees.

Important conversions include:

Worked Example

Evaluate:

sin(5π/6)

Convert to degrees:

5π/6 = 150°

The reference angle is 30°.

Sine is positive in Quadrant II.

Therefore:

sin(5π/6) = 1/2

Worked Example

Evaluate:

cot(7π/4)

Convert:

7π/4 = 315°

The reference angle is 45°.

Cotangent is negative in Quadrant IV.

Therefore:

cot(7π/4) = −1

When using a calculator, make sure it is set to the correct angle mode.

Evaluating Trigonometric Expressions

A trigonometric expression may contain several functions combined using arithmetic operations.

Worked Example: sin 30° + cos 60°

sin 30° = 1/2

cos 60° = 1/2

Therefore:

sin 30° + cos 60° = 1/2 + 1/2

= 1

Worked Example: 2 sin 45° cos 45°

Substitute:

2(√2/2)(√2/2)

= 2(2/4)

= 1

Worked Example: tan 60° × cot 60°

tan 60° = √3

cot 60° = √3/3

Therefore:

√3 × √3/3 = 3/3

= 1

This follows from the reciprocal relationship between tangent and cotangent.

Worked Example: sec² 60° − tan² 60°

sec 60° = 2

tan 60° = √3

Therefore:

sec² 60° − tan² 60°

= 2² − (√3)²

= 4 − 3

= 1

This is an example of a Pythagorean trigonometric identity.

Important Trigonometric Identities

A trigonometric identity is an equation that is true for every angle where both sides are defined.

Reciprocal Identities

csc θ = 1/sin θ

sec θ = 1/cos θ

cot θ = 1/tan θ

Quotient Identities

tan θ = sin θ/cos θ

cot θ = cos θ/sin θ

Pythagorean Identities

The fundamental identity is:

sin² θ + cos² θ = 1

Dividing by cos² θ gives:

1 + tan² θ = sec² θ

Dividing by sin² θ gives:

1 + cot² θ = csc² θ

These identities are valid wherever the expressions involved are defined.

Example

If:

tan θ = 2

find sec² θ.

Use:

1 + tan² θ = sec² θ

Substitute:

1 + 2² = sec² θ

sec² θ = 5

Even and Odd Trigonometric Functions

Trigonometric functions have useful symmetry properties.

An even function satisfies:

f(−θ) = f(θ)

An odd function satisfies:

f(−θ) = −f(θ)

Even Trigonometric Functions

Cosine and secant are even.

cos(−θ) = cos θ

sec(−θ) = sec θ

Odd Trigonometric Functions

Sine, tangent, cosecant, and cotangent are odd.

sin(−θ) = −sin θ

tan(−θ) = −tan θ

csc(−θ) = −csc θ

cot(−θ) = −cot θ

These relationships are useful when simplifying expressions involving negative angles.

Worked Example

Evaluate:

sin(−30°) + cos(−60°)

sin(−30°) = −1/2

cos(−60°) = 1/2

Therefore:

sin(−30°) + cos(−60°) = 0

Periodicity of Trigonometric Functions

A periodic function repeats its values after a fixed interval.

For sine and cosine:

sin(θ + 360°) = sin θ

cos(θ + 360°) = cos θ

For tangent and cotangent:

tan(θ + 180°) = tan θ

cot(θ + 180°) = cot θ

Cosecant and secant repeat every 360°.

Worked Example

Evaluate:

tan 585°

Since tangent has period 180°:

585° − 3(180°) = 45°

Therefore:

tan 585° = tan 45°

tan 585° = 1

Using periodicity often makes evaluating large angles much easier.

Finding Trigonometric Values from Coordinates

If a point (x, y) lies on the terminal side of an angle, its distance from the origin is:

r = √(x² + y²)

The trigonometric functions can then be calculated using:

• sin θ = y/r

• cos θ = x/r

• tan θ = y/x

• csc θ = r/y

• sec θ = r/x

• cot θ = x/y

These formulas apply whenever the denominators are nonzero.

Worked Example

A point on the terminal side of θ is:

(−3, 4)

Calculate the radius:

r = √[(-3)² + 4²]

r = √25

r = 5

Therefore:

sin θ = 4/5

cos θ = −3/5

tan θ = −4/3

csc θ = 5/4

sec θ = −5/3

cot θ = −3/4

The point lies in Quadrant II, which confirms the signs.

Applications of Trigonometric Functions

Trigonometric functions are used whenever quantities depend on angles or repeat periodically.

Physics: Resolving Forces

A force acting at an angle can be divided into horizontal and vertical components.

Fₓ = F cos θ

Fᵧ = F sin θ

Suppose a force of 80 N acts at 30° above the horizontal.

Horizontal component:

Fₓ = 80 cos 30°

Fₓ = 80(√3/2)

Fₓ = 40√3 N ≈ 69.3 N

Vertical component:

Fᵧ = 80 sin 30°

Fᵧ = 80(1/2)

Fᵧ = 40 N

Engineering: Circular Motion

The position of a point moving around a circle of radius r can be modelled using:

x = r cos θ

y = r sin θ

If θ changes with time, these become functions describing motion.

Waves and Oscillations

Sine and cosine functions model periodic motion.

For example:

y(t) = A sin(ωt + φ)

where:

• A = amplitude

• ω = angular frequency

• φ = phase constant

• t = time

This type of equation is used to describe sound waves, mechanical vibrations, and alternating electrical signals.

Navigation

Trigonometric functions allow distances and directions to be calculated.

For example, a displacement of 100 km at 60° north of east has components:

Eastward displacement:

100 cos 60° = 50 km

Northward displacement:

100 sin 60° = 50√3 km

Approximately:

86.6 km north

These calculations are useful in navigation, surveying, and engineering.

A Strategy for Evaluating Trigonometric Expressions

When evaluating a trigonometric expression:

1. Identify the trigonometric function.

2. Determine whether the angle is in degrees or radians.

3. Reduce the angle using periodicity if necessary.

4. Identify the quadrant.

5. Find the reference angle.

6. Determine the correct sign.

7. Use the exact special-angle value.

8. Apply reciprocal or quotient relationships if needed.

9. Simplify the expression.

10. Check whether the function is defined.

Example

Evaluate:

2 sec 120° − tan 225°

First:

cos 120° = −1/2

Therefore:

sec 120° = −2

Next:

tan 225° = 1

Substitute:

2(−2) − 1

= −4 − 1

= −5

Common Mistakes

Confusing sine and cosine

On the unit circle, cosine is the x-coordinate and sine is the y-coordinate.

Forgetting reciprocal relationships

Secant is the reciprocal of cosine, not sine. Cosecant is the reciprocal of sine.

Using the wrong quadrant sign

Always identify the quadrant before evaluating a trigonometric function.

Assuming all functions have a range from −1 to 1

Only sine and cosine have this range.

Ignoring undefined values

Tangent and secant are undefined when cosine is zero. Cotangent and cosecant are undefined when sine is zero.

Confusing degrees and radians

Check the angle mode on a calculator.

Assuming all trigonometric functions have the same period

Sine, cosine, cosecant, and secant have period 360°. Tangent and cotangent have period 180°.

Incorrectly squaring trigonometric functions

sin² θ means (sin θ)², not sin(θ²).

Rounding exact values unnecessarily

Use fractions and radicals when exact values are requested.

Applying identities where expressions are undefined

A trigonometric identity is valid only where both sides are defined.

Key Terms

Trigonometric Function: A function relating an angle to a numerical value based on geometric or unit-circle relationships.

Sine: The ratio opposite/hypotenuse in a right triangle, or the y-coordinate on the unit circle.

Cosine: The ratio adjacent/hypotenuse in a right triangle, or the x-coordinate on the unit circle.

Tangent: The ratio opposite/adjacent, equivalent to sine divided by cosine.

Cosecant: The reciprocal of sine.

Secant: The reciprocal of cosine.

Cotangent: The reciprocal of tangent.

Unit Circle: A circle of radius 1 centred at the origin.

Domain: The set of permitted input values of a function.

Range: The set of possible output values.

Quadrant: One of the four regions of the Cartesian coordinate plane.

Reference Angle: The acute angle formed between an angle's terminal side and the x-axis.

Coterminal Angles: Angles sharing the same terminal side.

Exact Value: A value expressed without rounding, often using fractions and radicals.

Reciprocal Identity: An identity relating a trigonometric function to its reciprocal.

Quotient Identity: An identity expressing tangent or cotangent as a ratio of sine and cosine.

Pythagorean Identity: An identity derived from sin² θ + cos² θ = 1.

Trigonometric Identity: An equation involving trigonometric functions that is true wherever both sides are defined.

Undefined Value: An input for which a function has no defined output.

Vertical Asymptote: A vertical line that a graph approaches as function values increase or decrease without bound.

Periodic Function: A function whose values repeat at regular intervals.

Period: The length of one complete repeating cycle.

Even Function: A function satisfying f(−x) = f(x).

Odd Function: A function satisfying f(−x) = −f(x).

Amplitude: The maximum displacement of an oscillation from its midline.

Angular Frequency: The rate of change of angular position, usually measured in radians per second.

Vector Component: The portion of a vector acting along a specified coordinate direction.

Check Your Understanding

1. Define all six trigonometric functions using right-triangle ratios.

2. Explain how sine and cosine are represented on the unit circle.

3. State the domain and range of the sine function.

4. Explain why tangent is undefined at 90°.

5. Determine the exact values of sin 60°, cos 30°, and tan 45°.

6. Evaluate csc 30°, sec 60°, and cot 45°.

7. Determine the signs of all six trigonometric functions in Quadrant III.

8. Find sin 150° and cos 150°.

9. Evaluate tan 225° and cot 315°.

10. Find sec(−120°).

11. Evaluate sin(7π/6) and cos(5π/3).

12. Simplify:

2 sin 30° cos 60° + tan 45°

13. If cos θ = −5/13 and θ lies in Quadrant II, determine sin θ and tan θ.

14. A point (−8, −6) lies on the terminal side of an angle θ. Determine all six trigonometric functions.

15. A force of 120 N acts at an angle of 135° measured counterclockwise from the positive x-axis. Calculate its horizontal and vertical components, giving exact values and decimal approximations.

Key Takeaways

• There are six trigonometric functions: sine, cosine, tangent, cosecant, secant, and cotangent.

• Sine, cosine, and tangent are the three primary trigonometric functions.

• Cosecant, secant, and cotangent are reciprocal functions.

• On the unit circle, cosine is the x-coordinate and sine is the y-coordinate.

• Tangent is the ratio of sine to cosine.

• The domain of sine and cosine is all real numbers.

• The range of sine and cosine is from −1 to 1.

• Tangent and cotangent have a range of all real numbers.

• Cosecant and secant have values less than or equal to −1 or greater than or equal to 1.

• Tangent and secant are undefined where cosine equals zero.

• Cotangent and cosecant are undefined where sine equals zero.

• The signs of trigonometric functions depend on the quadrant.

• Reference angles allow exact values to be determined for angles in any quadrant.

• Coterminal angles share the same trigonometric values.

• Sine, cosine, cosecant, and secant repeat every 360° or 2π radians.

• Tangent and cotangent repeat every 180° or π radians.

• Trigonometric identities help simplify and evaluate expressions.

• The fundamental Pythagorean identity is sin² θ + cos² θ = 1.

• Cosine and secant are even functions.

• Sine, tangent, cosecant, and cotangent are odd functions.

• Trigonometric functions are essential for analysing circular motion, waves, oscillations, navigation, and forces.

• Understanding the definitions, domains, ranges, signs, and relationships of all six trigonometric functions provides a strong foundation for advanced trigonometry and calculus.