Definitions of Sine, Cosine and Tangent and Exact Values — Edexcel A-level Mathematics (Pure)

SPEC 5.1 • TRIGONOMETRY

Definitions of Sine, Cosine and Tangent & Exact Values

Understanding trigonometric ratios for all arguments and memorising exact values

Learning Objectives

By the end of this lesson you will be able to:

Understand sine, cosine and tangent from right-angled triangles (SOHCAHTOA)

Define sin θ, cos θ, and tan θ for all angles using the unit circle

Determine the signs of trig functions in each quadrant using CAST

Work with negative angles and angles greater than 360°

Know and use exact values for 0°, 30°, 45°, 60°, 90° and their multiples

Right-Angled Triangle Definitions

In a right-angled triangle, the trigonometric ratios are defined as:

SOHCAHTOA

sin θ = \frac{Opposite}{Hypotenuse}

cos θ = \frac{Adjacent}{Hypotenuse}

tan θ = \frac{Opposite}{Adjacent}

Remember

SOH - Sine = Opposite / Hypotenuse

CAH - Cosine = Adjacent / Hypotenuse

TOA - Tangent = Opposite / Adjacent

Limitations of Triangle Definitions

Problem

Right-angled triangle definitions only work for angles between 0° and 90°.

But we need trigonometric functions that work for:

Angles greater than 90° (e.g., 150°, 270°)

Negative angles (e.g., -45°, -120°)

Angles greater than 360° (e.g., 450°, 720°)

Solution

We use the unit circle to extend these definitions to all angles.

The Unit Circle

Definition

The unit circle is a circle with centre at the origin (0, 0) and radius 1.

Its equation is:

x^{2} + y^{2} = 1

Coordinates on the Unit Circle

For any angle θ measured anticlockwise from the positive x-axis:

Key ResultPoint P on unit circle:

P = (cos θ, sin θ)

Definitions for All Angles

Unit Circle Definitions

cos θ

= the x-coordinate of P

sin θ

= the y-coordinate of P

tan θ = \frac{s i n θ}{c o s θ}

= the gradient of OP

Important

These definitions work for any angle — positive, negative, or greater than 360°.

Since P lies on the unit circle:

- 1 \leq cos θ \leq 1

and

- 1 \leq sin θ \leq 1

Finding Values from the Unit Circle

Worked Example01

Using the unit circle, state the values of:

(a) sin 90°

(b) cos 180°

(c) tan 270°

Part (a) At θ = 90°, P = (0, 1)

sin 90° = y-coordinate = 1

Part (b) At θ = 180°, P = (-1, 0)

cos 180° = x-coordinate = -1

Part (c) At θ = 270°, P = (0, -1)

tan 270° = sin 270° / cos 270° = -1 / 0 = undefined

The Four Quadrants

The coordinate axes divide the plane into four quadrants:

QuadrantAngle Rangex-coord (cos θ)y-coord (sin θ)

1st0° to 90°Positive (+)Positive (+)

2nd90° to 180°Negative (−)Positive (+)

3rd180° to 270°Negative (−)Negative (−)

4th270° to 360°Positive (+)Negative (−)

The CAST Diagram

The CAST diagram tells us which trig functions are positive in each quadrant:

SSin positive

AAll positive

TTan positive

CCos positive

Memory Aid

Read anticlockwise from Q4: C-A-S-T

Or remember: "All Students Take Calculus" (starting from Q1)

Signs of Trig Functions

Quadrantsin θcos θtan θ

1st (A)+++

2nd (S)+−−

3rd (T)−−+

4th (C)−+−

Why?

tan θ = sin θ / cos θ, so tan is positive when sin and cos have the same sign.

Using the CAST Diagram

Worked Example02

State whether each of the following is positive or negative:

(a) sin 140°

(b) cos 250°

(c) tan 320°

Part (a) 140° is in the 2nd quadrant (90° < 140° < 180°)

In Q2, only sin is positive. So sin 140° is positive.

Part (b) 250° is in the 3rd quadrant (180° < 250° < 270°)

In Q3, only tan is positive. So cos 250° is negative.

Part (c) 320° is in the 4th quadrant (270° < 320° < 360°)

In Q4, only cos is positive. So tan 320° is negative.

Negative Angles

Definition

A negative angle is measured clockwise from the positive x-axis.

Properties of Negative Angles

sin (- θ) = - sin θ

— sine is an odd function

cos (- θ) = cos θ

— cosine is an even function

tan (- θ) = - tan θ

— tangent is an odd function

Angles Greater Than 360°

Coterminal Angles

Angles that differ by multiples of 360° are called coterminal — they have the same terminal side on the unit circle.

Method

To evaluate trig functions for angles > 360°:

Subtract 360° (repeatedly if needed) until you get an angle between 0° and 360°.

Examplessin 450° = sin(450° − 360°) = sin 90° = 1 cos 720° = cos(720° − 360° − 360°) = cos 0° = 1

Exact Trigonometric Values

Important

You must memorise the exact values of sin, cos, and tan for these angles:

0°, 30°, 45°, 60°, 90°

These are given in the formula booklet, but knowing them speeds up your work significantly.

Why These Angles?

These come from two special triangles: the 45-45-90 triangle and the 30-60-90 triangle.

The 45-45-90 Triangle

An isosceles right-angled triangle with two 45° angles:

Exact Values for 45°

sin 45° = \frac{1}{2} = \frac{2}{2}

cos 45° = \frac{1}{2} = \frac{2}{2}

tan 45° = \frac{1}{1} = 1

The 30-60-90 Triangle

Half of an equilateral triangle with side length 2:

Key Sides

Shortest side = 1 (opposite 30°), Medium = √3 (opposite 60°), Hypotenuse = 2

Exact Values for 30° and 60°

For 30°

sin 30° = \frac{1}{2}

cos 30° = \frac{3}{2}

tan 30° = \frac{1}{3} = \frac{3}{3}

For 60°

sin 60° = \frac{3}{2}

cos 60° = \frac{1}{2}

tan 60° = 3

Pattern

Notice: sin 30° = cos 60° and sin 60° = cos 30° (complementary angles)

Complete Exact Values Table

θ0°30°45°60°90°

sin θ0

\frac{1}{2}

\frac{2}{2}

\frac{3}{2}

1

cos θ1

\frac{3}{2}

\frac{2}{2}

\frac{1}{2}

0

tan θ0

\frac{3}{3}

1

3

undef

Memory Trick for Sin

\frac{0}{2}, \frac{1}{2}, \frac{2}{2}, \frac{3}{2}, \frac{4}{2}

gives 0, ½,

\frac{2}{2}

,

\frac{3}{2}

, 1

Using Exact Values

Worked Example04

Find the exact value of:

(a) sin 150°

(b) cos 240°

(c) tan 315°

Part (a) 150° is in Q2, related angle = 180° − 150° = 30°

In Q2, sin is positive.

sin 150° = +sin 30° = ½

Part (b) 240° is in Q3, related angle = 240° − 180° = 60°

In Q3, cos is negative.

cos 240° = −cos 60° = −½

Part (c) 315° is in Q4, related angle = 360° − 315° = 45°

In Q4, tan is negative.

tan 315° = −tan 45° = −1

Exact Values with Special Angles

Worked Example05

Find the exact value of:

(a) sin(−60°)

(b) cos 420°

(c) tan(−135°)

Part (a) sin(−60°) = −sin 60° = $- \frac{3}{2}$

Part (b) 420° − 360° = 60°

cos 420° = cos 60° = ½

Part (c) tan(−135°) = −tan 135°

135° is in Q2, related angle = 45°, tan is negative in Q2

tan 135° = −tan 45° = −1

So tan(−135°) = −(−1) = 1

Evaluating Expressions

Worked Example06

Find the exact value of:

sin^{2} 60° + cos^{2} 60°

Step 1 Substitute exact values:

sin 60° = \frac{3}{2}

,

cos 60° = \frac{1}{2}

Step 2 Calculate:

sin^{2} 60° + cos^{2} 60° = (\frac{3}{2})^{2} + (\frac{1}{2})^{2}

= \frac{3}{4} + \frac{1}{4} = \frac{4}{4} = 1

Note: This is actually true for ALL angles — it's the Pythagorean identity!

Exact Values in Radians

Later you'll learn radians. The same exact values apply:

Degrees0°30°45°60°90°

Radians0

\frac{π}{6}

\frac{π}{4}

\frac{π}{3}

\frac{π}{2}

For Now

Focus on memorising the degree values. The radian equivalents will come in Lesson 5.03.

Practice

1 State the quadrant and sign of:

(a) sin 200°

(b) cos 340°

(c) tan 95°

Part (a) 200° is in Q3 (180° < 200° < 270°)

In Q3, only tan is positive. sin 200° is negative.

Part (b) 340° is in Q4 (270° < 340° < 360°)

In Q4, only cos is positive. cos 340° is positive.

Part (c) 95° is in Q2 (90° < 95° < 180°)

In Q2, only sin is positive. tan 95° is negative.

Practice

2 Find the exact value of:

(a) sin 330°

(b) cos 225°

(c) tan 120°

Part (a) 330° is in Q4, related angle = 360° − 330° = 30°

In Q4, sin is negative.

sin 330° = −sin 30° = −½

Part (b) 225° is in Q3, related angle = 225° − 180° = 45°

In Q3, cos is negative.

cos 225° = −cos 45° = $- \frac{2}{2}$

Part (c) 120° is in Q2, related angle = 180° − 120° = 60°

In Q2, tan is negative.

tan 120° = −tan 60° = −√3

Learning Objectives

Right-Angled Triangle Definitions

Limitations of Triangle Definitions

The Unit Circle

Coordinates on the Unit Circle

Definitions for All Angles

Finding Values from the Unit Circle

The Four Quadrants

The CAST Diagram

SSin positive

AAll positive

TTan positive

CCos positive

Signs of Trig Functions

Using the CAST Diagram

Related Acute Angles

Finding Related Acute Angles

Negative Angles

Angles Greater Than 360°

Exact Trigonometric Values

The 45-45-90 Triangle

The 30-60-90 Triangle

Exact Values for 30° and 60°

Complete Exact Values Table

Using Exact Values

Exact Values with Special Angles

Evaluating Expressions

Exact Values in Radians

Practice

Practice