D2.3 Connect sine and cosine ratios with sine and cosine functions

Ontario Grade 11 Mathematics

Trigonometric Functions

Connecting Triangle Ratios to the Sine and Cosine Functions (MCR3U – D2.3)

You already know how to find the sine and cosine of an acute angle using a right triangle. In this lesson you will see how that same idea stretches far beyond triangles to produce two of the most important functions in mathematics. By the end, you will understand why a ratio you first met in a right triangle can produce a smooth, repeating wave that describes everything from sound to tides.

What you will learn

Explain what a sine or cosine ratio means in a right triangle and on the unit circle.
Describe how the sine and cosine ratios for any angle become the sine and cosine functions.
Read and interpret a graph of y = sin θ or y = cos θ, identifying key features such as period, amplitude, and the behaviour at benchmark angles.
Use known triangle ratios to verify values on a sine or cosine graph.
Evaluate and compare sine and cosine values for angles in all four quadrants using the CAST rule.

Prerequisite Bridge: Sine and Cosine as Ratios

In Grade 10 you defined the primary trigonometric ratios for an acute angle θ in a right triangle. If the side opposite θ has length opp, the side adjacent to θ has length adj, and the hypotenuse has length hyp, then the two ratios are

\sin\theta = \frac{\text{opp}}{\text{hyp}}

and

\cos\theta = \frac{\text{adj}}{\text{hyp}}

.

These ratios are pure numbers — they have no units. Because the hypotenuse is always the longest side of a right triangle, both ratios are always between −1 and 1. For an acute angle, both ratios are positive. This range will become the amplitude of the functions you are about to meet.

One important set of values to remember: for

\theta = 30°

,

\sin 30° = 0.5

and

\cos 30° \approx 0.866

; for

\theta = 45°

,

\sin 45° = \cos 45° \approx 0.707

; for

\theta = 60°

,

\sin 60° \approx 0.866

and

\cos 60° = 0.5

. These exact values come directly from the special triangles (30-60-90 and 45-45-90) you studied in Grade 10.

\sin\theta = \frac{\text{opp}}{\text{hyp}}, \cos\theta = \frac{\text{adj}}{\text{hyp}}

$\sin\theta = \frac{\text{opp}}{\text{hyp}}$ and $\cos\theta = \frac{\text{adj}}{\text{hyp}}$ for any acute angle in a right triangle.
Both ratios are always in the interval from −1 to 1.
Benchmark values at 0°, 30°, 45°, 60°, and 90° should be memorised or derivable from special triangles.

Extending the Ratios Beyond 90°: The Unit Circle Idea

A right triangle cannot have an angle greater than 90°, so the ratio definitions above only work for acute angles. To give sine and cosine a meaning for any angle — including obtuse and reflex angles — mathematicians place a point P on a circle of radius 1 centred at the origin. This circle is called the unit circle.

Imagine an angle θ measured from the positive x-axis, rotating counter-clockwise. The point P where the terminal arm meets the unit circle has coordinates

(\cos\theta,\, \sin\theta)

. This is the key connection: the x-coordinate of P equals

\cos\theta

and the y-coordinate of P equals

\sin\theta

, for any angle θ, not just acute ones.

Why does this match the right-triangle definition? For an acute angle, draw a vertical line from P down to the x-axis. You get a right triangle whose hypotenuse is the radius 1, whose vertical leg is the y-coordinate (opposite side), and whose horizontal leg is the x-coordinate (adjacent side). So

\sin\theta = y/1 = y

and

\cos\theta = x/1 = x

, which matches the coordinates perfectly.

For angles beyond 90°, the x- or y-coordinate of P may be negative, which is exactly why sine and cosine can be negative. The ratio definition and the coordinate definition agree completely for acute angles, and the coordinate definition smoothly extends the idea to all angles.

P = (\cos\theta, \sin\theta)

Place angle θ on a circle of radius 1; the terminal-arm endpoint is $(\cos\theta, \sin\theta)$ .
This extends sine and cosine to angles of any size, including those greater than 90° or less than 0°.
For acute angles, the unit-circle definition gives the same result as the right-triangle ratio.
The sign of sine or cosine depends on which quadrant the terminal arm lands in.

The CAST Rule: Signs in Each Quadrant

Because the unit circle extends into all four quadrants, the signs of sine and cosine change depending on where the terminal arm points. The CAST rule is a quick memory tool. Reading counter-clockwise from the fourth quadrant: C (Cosine positive), A (All positive), S (Sine positive), T (Tangent positive). For this lesson, focus on the S and C parts.

In Quadrant I (0° to 90°), both

\sin\theta > 0

and

\cos\theta > 0

. In Quadrant II (90° to 180°),

\sin\theta > 0

but

\cos\theta < 0

. In Quadrant III (180° to 270°), both are negative. In Quadrant IV (270° to 360°),

\cos\theta > 0

but

\sin\theta < 0

.

The reference angle is the acute angle between the terminal arm and the x-axis. You find the exact numerical value of

\sin\theta

or

\cos\theta

using the reference angle and then apply the correct sign from the CAST rule. For example, the reference angle for 150° is 30°, so

|\sin 150°| = \sin 30° = 0.5

and since 150° is in Quadrant II where sine is positive,

\sin 150° = 0.5

.

CAST: Quadrant I — All positive; Quadrant II — Sine positive; Quadrant III — Tangent positive; Quadrant IV — Cosine positive.
Find the value using the reference angle, then assign the correct sign using CAST.
The reference angle is always acute (between 0° and 90°).

From Ratio to Function: Building the Graph

A function is a rule that assigns exactly one output to every input. The sine function takes an angle θ as input and returns the number

\sin\theta

as output. Written in function notation:

f(\theta) = \sin\theta

. The cosine function is

g(\theta) = \cos\theta

. Both functions accept any real angle in degrees.

To see what the graph looks like, calculate

\sin\theta

for angles from 0° to 360° and plot the points

(\theta,\, \sin\theta)

. At

\theta = 0°

,

\sin 0° = 0

. At

\theta = 90°

,

\sin 90° = 1

. At

\theta = 180°

,

\sin 180° = 0

. At

\theta = 270°

,

\sin 270° = -1

. At

\theta = 360°

,

\sin 360° = 0

. Connecting these with a smooth curve produces one complete wave — called one cycle.

The graph of

y = \cos\theta

follows the same logic. At

\theta = 0°

,

\cos 0° = 1

. At

\theta = 90°

,

\cos 90° = 0

. At

\theta = 180°

,

\cos 180° = -1

. At

\theta = 270°

,

\cos 270° = 0

. At

\theta = 360°

,

\cos 360° = 1

. Notice that the cosine graph has the same shape as the sine graph but starts at 1 instead of 0 — it is shifted 90° to the left.

Both graphs repeat perfectly every 360°. This repeating behaviour is called periodicity. The period of both

y = \sin\theta

and

y = \cos\theta

is 360°. The amplitude is 1 because the graph reaches a maximum of 1 and a minimum of −1, and the amplitude is half the total vertical range:

(1 - (-1)) / 2 = 1

.

\text{Amplitude} = 1, \text{Period} = 360°

The sine and cosine functions assign one output value to every input angle θ.
One full cycle of $y = \sin\theta$ or $y = \cos\theta$ spans 360°.
Both graphs have amplitude 1 and period 360°.
The cosine graph is the sine graph shifted 90° to the left: $\cos\theta = \sin(\theta + 90°)$ .
Every point on the graph is derived directly from the unit-circle coordinates.

Key Features and How the Ratio Confirms the Function

The maximum value of

y = \sin\theta

is 1, reached at

\theta = 90°

. The minimum is −1, reached at

\theta = 270°

. The graph crosses the x-axis (called x-intercepts or zeros) at CAD 0°, 180°, 360°, and every 180° beyond that. These zeros correspond directly to the angles where the opposite side in a right triangle has length 0 relative to the hypotenuse — that is, when the terminal arm is along the x-axis.

For

y = \cos\theta

, the maximum is 1 at

\theta = 0°

and CAD 360°, the minimum is −1 at

\theta = 180°

, and the zeros are at CAD 90° and CAD 270°. These zeros correspond to angles where the adjacent side has length 0 — that is, when the terminal arm is vertical.

This direct connection between the triangle-ratio meaning and the function graph is what the expectation D2.3 asks you to understand. When you read

\sin 150° = 0.5

from a graph, you can confirm it by finding the reference angle (30°), using the special triangle to get

\sin 30° = 0.5

, and checking that 150° is in Quadrant II where sine is positive. The graph, the ratio, and the CAST rule all agree.

Zeros of $y = \sin\theta$ occur where the terminal arm is on the x-axis; zeros of $y = \cos\theta$ occur where it is on the y-axis.
A value read from the graph can always be verified using a reference angle and the CAST rule.
The connection between ratio and function is exact, not approximate.

Benchmark Values of Sine and Cosine (0° to 360°)

Angle θ	sin θ	cos θ	Quadrant
0°	0	1	— (on axis)
30°	0.500	0.866	I
45°	0.707	0.707	I
60°	0.866	0.500	I
90°	1	0	— (on axis)
120°	0.866	−0.500	II
150°	0.500	−0.866	II
180°	0	−1	— (on axis)

Worked example

Example 1 – Linking a Triangle Ratio to a Point on the Sine Graph

A right triangle has an angle of 40° and a hypotenuse of 1. Show that the side opposite the 40° angle equals sin 40°, then explain what this tells you about the point on the graph of y = sin θ at θ = 40°.

Write the ratio definition
Start with the ratio definition: $\sin\theta = \frac{\text{opp}}{\text{hyp}}$ . Here $\theta = 40°$ and the hypotenuse equals 1.
$\sin 40° = \frac{\text{opp}}{1}$
Solve for the opposite side
Multiply both sides by 1. The opposite side has exactly the same numerical value as $\sin 40°$ .
$\text{opp} = \sin 40° \approx 0.643$
Connect to the unit circle
When the hypotenuse is 1, the triangle fits perfectly inside the unit circle. The endpoint of the hypotenuse is the point $P = (\cos 40°, \sin 40°)$ . The y-coordinate of P is the opposite side, which equals $\sin 40°$ .
$P = (\cos 40°, \sin 40°) \approx (0.766, 0.643)$
Read the graph
On the graph of $y = \sin\theta$ , locate $\theta = 40°$ on the horizontal axis. The height of the graph at that point is the y-coordinate we just found, approximately 0.643. The ratio and the function value are the same number.
$y = \sin 40° \approx 0.643$

Answer: The side opposite the 40° angle in a right triangle with hypotenuse 1 has length approximately 0.643, which equals sin 40°. The point on the graph of y = sin θ at θ = 40° is therefore (40°, 0.643). The triangle ratio and the function value are identical.

Check: Using a calculator: sin 40° ≈ 0.6428. The opposite side in the unit-circle triangle is also ≈ 0.6428. Both values match, confirming the connection.

Worked example

Example 2 – Evaluating Sine and Cosine for an Obtuse Angle Using Both Methods

Find sin 120° and cos 120° by (a) using the reference angle and the CAST rule, and (b) confirming the values match what a correctly drawn graph of y = sin θ and y = cos θ would show at θ = 120°.

Identify the quadrant and reference angle
An angle of 120° lies in Quadrant II because it is between 90° and 180°. The reference angle is found by subtracting from 180°: $180° - 120° = 60°$ . So the reference angle is 60°.
$\theta_{\text{ref}} = 180° - 120° = 60°$
Find the numerical values using the 30-60-90 triangle
From the special 30-60-90 triangle, $\sin 60° = \frac{\sqrt{3}}{2}$ and $\cos 60° = \frac{1}{2}$ . These are the magnitudes.
$\sin 60° = \frac{\sqrt{3}}{2} \approx 0.866, \cos 60° = \frac{1}{2} = 0.5$
Apply the CAST rule for Quadrant II
In Quadrant II, the CAST rule tells us sine is positive and cosine is negative. Apply the correct signs to the magnitudes found in the previous step.
$\sin 120° = +\frac{\sqrt{3}}{2} \approx 0.866, \cos 120° = -\frac{1}{2} = -0.5$
Confirm with the unit-circle coordinates
At 120°, the terminal arm endpoint is $P = (\cos 120°, \sin 120°) = (-0.5,\; 0.866)$ . The x-coordinate is negative (Quadrant II has negative x-values) and the y-coordinate is positive (Quadrant II has positive y-values). This matches our CAST result.
$P = (-0.5, 0.866)$
Read the graphs
On the graph of $y = \sin\theta$ , the height at $\theta = 120°$ is approximately 0.866 — the graph is still above the x-axis, which makes sense because 120° is in Quadrant II where sine is positive. On the graph of $y = \cos\theta$ , the height at $\theta = 120°$ is −0.5 — the graph is below the x-axis, consistent with cosine being negative in Quadrant II.

Answer: sin 120° = √3/2 ≈ 0.866 and cos 120° = −1/2 = −0.5. Both values can be read from the respective graphs at θ = 120° and confirmed using the reference angle 60° and the CAST rule.

Check: Calculator check: sin 120° ≈ 0.8660 and cos 120° = −0.5000. These match the exact values √3/2 and −1/2 respectively. No errors found.

Common mistakes and how to avoid them

Thinking sine and cosine are undefined for angles greater than 90° because a right triangle cannot have such an angle.

Correction: The unit-circle definition extends sine and cosine to all angles. The right-triangle definition is just the starting point for acute angles.

Forgetting to apply the correct sign from the CAST rule and writing, for example, sin 150° = −0.5 instead of +0.5.

Correction: Always identify the quadrant first, then find the magnitude using the reference angle, and finally assign the sign using CAST. In Quadrant II, sine is positive.

Confusing the period (360°) with the amplitude (1). Some learners say the graph 'repeats with amplitude 360.'

Correction: Period is the horizontal distance for one complete cycle (360°). Amplitude is the maximum distance from the midline to the peak (1). They measure different things.

Reading the cosine graph as though it starts at 0 the way the sine graph does.

Correction: The cosine graph starts at its maximum value of 1 when θ = 0°, while the sine graph starts at 0. Always check which function you are reading.

Assuming cos θ = sin θ for all angles because they are equal at 45°.

Correction: sin θ = cos θ only at θ = 45° and θ = 225° within one cycle. For all other angles the values differ — confirm with the benchmark table.

Lesson summary

The sine and cosine ratios from right triangles — opposite over hypotenuse and adjacent over hypotenuse — are the foundation for the sine and cosine functions.
Placing the angle on the unit circle (a circle of radius 1) extends these ratios to any angle by defining sin θ and cos θ as the y- and x-coordinates of the terminal-arm endpoint.
The CAST rule gives the sign of each ratio in each quadrant; the magnitude always comes from the reference angle.
Plotting (θ, sin θ) and (θ, cos θ) for all angles produces smooth, repeating graphs, each with amplitude 1 and period 360°.
Every value on a sine or cosine graph can be verified using the matching ratio: find the reference angle, use a special triangle or calculator for the magnitude, then apply the CAST sign.
The cosine graph has the same wave shape as the sine graph but is shifted 90° to the left, starting at its maximum value of 1 at θ = 0°.

Check your understanding

Question 1

A right triangle has hypotenuse 1 and an angle of 55°. What is the length of the side opposite the 55° angle?

cos 55° ≈ 0.574
sin 55° ≈ 0.819
tan 55° ≈ 1.428
1 ÷ sin 55° ≈ 1.221

Show answer and explanation

sin 55° ≈ 0.819

Using sin θ = opp/hyp with hyp = 1: opp = sin 55° × 1 = sin 55° ≈ 0.819. This is also the y-coordinate of the terminal-arm point on the unit circle at 55°.

Question 2

What is sin 210°? Use the reference angle and CAST rule.

0.500
0.866
−0.500
−0.866

Show answer and explanation

−0.500

210° is in Quadrant III (between 180° and 270°). The reference angle is 210° − 180° = 30°. From the special triangle, sin 30° = 0.5. In Quadrant III, sine is negative, so sin 210° = −0.5.

Question 3

On the graph of y = cos θ, what is the y-value (output) when θ = 180°?

1
0
−1
0.5

Show answer and explanation

−1

At θ = 180°, the terminal arm points in the negative x-direction on the unit circle. The x-coordinate of that endpoint is −1, so cos 180° = −1. The graph of y = cos θ reaches its minimum of −1 at θ = 180°.

Question 4

Which statement correctly describes how the graph of y = cos θ relates to the graph of y = sin θ?

The cosine graph is a vertical reflection of the sine graph.
The cosine graph has a larger amplitude than the sine graph.
The cosine graph has the same shape as the sine graph but starts at its maximum value of 1 when θ = 0°.
The cosine graph has a period of 180° while the sine graph has a period of 360°.

Show answer and explanation

The cosine graph has the same shape as the sine graph but starts at its maximum value of 1 when θ = 0°.

Both graphs have the same wave shape and the same amplitude (1) and period (360°). The key difference is the starting value: sin 0° = 0, but cos 0° = 1. The cosine graph is the sine graph shifted 90° to the left.

Key terms

Sine ratio: In a right triangle, the ratio of the length of the side opposite an angle to the length of the hypotenuse.
Cosine ratio: In a right triangle, the ratio of the length of the side adjacent to an angle to the length of the hypotenuse.
Unit circle: A circle with radius 1 centred at the origin, used to define sine and cosine for angles of any size. The coordinates of the point where the terminal arm meets the circle are (cos θ, sin θ).
Terminal arm: The rotating ray that forms an angle θ when measured from the positive x-axis. Its endpoint on the unit circle gives the sine and cosine values for θ.
Reference angle: The acute angle (between 0° and 90°) between the terminal arm and the nearest part of the x-axis. It gives the magnitude of the sine or cosine value.
CAST rule: A memory tool for the signs of trigonometric ratios by quadrant. Counter-clockwise from Quadrant IV: Cosine positive, All positive, Sine positive, Tangent positive.
Amplitude: The maximum distance a graph reaches above or below its midline. For y = sin θ and y = cos θ, the amplitude is 1.
Period: The horizontal length of one complete cycle of a repeating graph. For y = sin θ and y = cos θ, the period is 360°.

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About this lesson

Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 11 Mathematics (MCR3U), expectation D2.3. It is a study resource, not an official curriculum publication.

Official curriculum reference

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D2.3 · Connect sine and cosine ratios with sine and cosine functions

Ontario Grade 11 Mathematics

Connecting Triangle Ratios to the Sine and Cosine Functions (MCR3U – D2.3)

What you will learn

Prerequisite Bridge: Sine and Cosine as Ratios

Extending the Ratios Beyond 90°: The Unit Circle Idea

The CAST Rule: Signs in Each Quadrant

From Ratio to Function: Building the Graph

Key Features and How the Ratio Confirms the Function

Benchmark Values of Sine and Cosine (0° to 360°)

Example 1 – Linking a Triangle Ratio to a Point on the Sine Graph

Example 2 – Evaluating Sine and Cosine for an Obtuse Angle Using Both Methods

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Question 4

Key terms

Continue through MCR3U

About this lesson