Connect the sine ratio with the sine function

Ontario Grade 11 Mathematics

Trigonometric Functions

Connecting a right-triangle ratio to angle inputs and sine outputs

In a right triangle, sine compares two side lengths. The sine function uses that familiar ratio to assign a value to an angle. We will connect the triangle, the unit circle, a table, and a graph. All angles in this lesson are measured in degrees.

What you will learn

Recall the sine ratio for an acute angle in a right triangle.
Explain how the sine function uses the same ratio for acute angles.
Interpret sine as a point’s vertical coordinate on a unit circle.
Connect sine values to a table and a graph.

1. Prerequisite bridge: the right-triangle sine ratio

A right triangle has one

90^\circ

angle. Choose one of its other angles. The side across from that chosen angle is the opposite side. The hypotenuse is the longest side and lies across from the

90^\circ

angle.

For an acute angle, the sine ratio is the length of the opposite side divided by the length of the hypotenuse. For example, if those side lengths are

3

and

5

, the ratio is

\frac{3}{5}

. The ratio depends on the angle, not on the overall size of the triangle. Triangles with the same angles have the same side-length ratios.

\sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}}

Choose the angle before naming its opposite side.
The right-triangle sine ratio applies directly to acute angles.

2. Plain-language connection: the unit circle

A unit circle is a circle with radius

1

centred at the origin of a coordinate grid. Imagine starting at

(1,0)

and turning counterclockwise through an angle. The point where the turn ends lies on the circle.

For an acute angle, the point is above the horizontal axis. A vertical line from the point to that axis makes a right triangle. The circle’s radius is the triangle’s hypotenuse, so its length is

1

. The vertical side is opposite the angle. The triangle ratio is therefore the vertical side divided by

1

, which equals the point’s vertical coordinate.

This explains the connection: for an acute angle, the right-triangle sine ratio equals the vertical coordinate of the point on the unit circle. The sine function assigns that value to the angle. A function is a rule that gives one output for each input.

The unit-circle view also lets us discuss angles whose points are not in the first quadrant. A point above the horizontal axis has a positive vertical coordinate; a point below it has a negative one. On the axis, the vertical coordinate is zero. Sine is still the vertical coordinate, so its output can be positive, zero, or negative.

\sin(\theta)=y

A unit circle has radius $1$ .
For an acute angle, the triangle ratio equals the unit-circle point’s vertical coordinate.
The sine function’s input is an angle and its output is a number.

3. Multiple representations: table and graph

The table lists points where the unit circle meets the axes. Since sine is the vertical coordinate, each point gives a sine value. These entries are useful checkpoints when reading or sketching a graph.

A graph of

y=\sin(\theta)

places angle inputs along the horizontal axis and sine outputs along the vertical axis. The graph passes through the table’s values. Between them, it rises and falls smoothly. Its outputs stay between

-1

and

1

, because the vertical coordinate of a point on a circle of radius

1

cannot be above

1

or below

-1

.

After a turn of

360^\circ

, the point returns to its starting location. The sine value therefore repeats. The graph repeats the same pattern every

360^\circ

. When using a calculator to check a value, make sure it is set to degrees if the angle includes a degree symbol.

\sin(\theta+360^\circ)=\sin(\theta)

The graph’s vertical coordinate is the sine value.
The sine values on the unit circle range from $-1$ to $1$ .
The sine pattern repeats after $360^\circ$ .

4. Applying the connection to another angle

A reference angle is the acute angle between the terminal arm of an angle and the horizontal axis. It helps connect an angle outside the first quadrant to a familiar acute-angle ratio.

The reference angle gives the size of the vertical coordinate. The position of the point tells whether that coordinate is positive or negative. For example, in the second quadrant the point is above the horizontal axis, so its sine value is positive. In the third and fourth quadrants it is below the axis, so the sine value is negative.

This method does not treat every angle as an angle inside a right triangle. Instead, it uses the unit-circle definition to extend the familiar sine ratio: the acute reference angle provides a familiar magnitude, and the point’s location provides the sign.

\sin(150^\circ)=\sin(30^\circ)

Use the reference angle to connect to an acute-angle sine ratio.
Use the point’s position above or below the horizontal axis to determine the sign.

Axis points on the unit circle

Angle	Point	Sine value
$0^\circ$	$(1,0)$	$0$
$90^\circ$	$(0,1)$	$1$
$180^\circ$	$(-1,0)$	$0$
$270^\circ$	$(0,-1)$	$-1$
$360^\circ$	$(1,0)$	$0$

Worked example

Finding sine for an obtuse angle

Determine the exact value of

\sin(150^\circ)

and explain how it connects to a right-triangle sine ratio.

Locate the angle
The angle ends in the second quadrant. Points there are above the horizontal axis, so their vertical coordinates, and therefore their sine values, are positive.
Find the reference angle
The reference angle is the acute angle between the terminal arm and the horizontal axis. Subtract the given angle from $180^\circ$ to find it.
$180^\circ-150^\circ=30^\circ$
Connect to the acute-angle ratio
The unit-circle point for $150^\circ$ has the same vertical coordinate as the point for $30^\circ$ . The right-triangle sine ratio for $30^\circ$ gives that coordinate as one-half. The value is positive in the second quadrant.
$\sin(150^\circ)=\sin(30^\circ)=\frac{1}{2}$

Answer:

\sin(150^\circ)=\frac{1}{2}

Check: The value is positive because the point is above the horizontal axis. Its size matches the sine ratio for the

30^\circ

reference angle.

Common mistakes and how to avoid them

Using the horizontal coordinate as the sine value.

Correction: Sine is the vertical coordinate. The horizontal coordinate is a different coordinate.

Assuming sine must always be positive because triangle side lengths are positive.

Correction: The ratio uses positive side lengths for acute triangle angles. The function also assigns values to points below the horizontal axis, where the vertical coordinate is negative.

Using a reference angle without checking the point’s position.

Correction: The reference angle gives the size of the sine value. Check whether the point is above or below the horizontal axis to choose its sign.

Checking a degree angle on a calculator set to a different angle mode.

Correction: Use degree mode when the angle is marked with

^\circ

.

Lesson summary

For an acute angle in a right triangle, sine is opposite divided by hypotenuse.
On the unit circle, sine is the vertical coordinate of the point reached by the angle.
For acute angles, the right-triangle ratio matches the unit-circle coordinate.
The sine graph shows angle inputs and their sine outputs, which range from $-1$ to $1$ .
A reference angle connects a broader angle to a familiar acute-angle ratio.

Check your understanding

Question 1

A point on the unit circle reached by

210^\circ

is below the horizontal axis. What must be true about

\sin(210^\circ)

?

It is negative because sine is the vertical coordinate.
It is positive because sine is a ratio of side lengths.
It is zero because the angle is greater than $180^\circ$ .
It is the horizontal coordinate of the point.

Show answer and explanation

It is negative because sine is the vertical coordinate.

Sine is the vertical coordinate. A point below the horizontal axis has a negative vertical coordinate.

Question 2

What is the reference angle for

240^\circ

?

$30^\circ$
$60^\circ$
$120^\circ$
$240^\circ$

Show answer and explanation

60^\circ

The angle is in the third quadrant. The reference angle is the difference between

240^\circ

and

180^\circ

, which is

60^\circ

.

Question 3

Which statement connects the right-triangle sine ratio to the sine function?

For an acute angle, the triangle ratio equals the unit-circle point’s vertical coordinate.
For every angle, sine is opposite divided by hypotenuse in a right triangle.
For an acute angle, sine is the unit-circle point’s horizontal coordinate.
The sine function only gives values from $0$ to $1$ .

Show answer and explanation

For an acute angle, the triangle ratio equals the unit-circle point’s vertical coordinate.

For an acute angle, the unit-circle radius is the hypotenuse of length

1

, and the opposite side is the vertical coordinate. Dividing by

1

gives that same coordinate.

Key terms

Sine ratio: For an acute angle in a right triangle, the opposite side length divided by the hypotenuse length.
Unit circle: A circle of radius $1$ centred at the origin of a coordinate grid.
Sine function: A rule that assigns an angle a sine value; on the unit circle, that value is the vertical coordinate.
Reference angle: The acute angle between an angle’s terminal arm and the horizontal axis.
Coordinate: One of the numbers that describes a point’s position on a coordinate grid.

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Published by DoAssignment. This reviewed lesson follows Ontario Grade 11 Mathematics (MCF3M), expectation C2.3. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

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