Trigonometric Functions on the Unit Circle: Videos & Practice Problems

The unit circle connects angles to coordinates using trigonometric functions: sine, cosine, and tangent. For any angle, sine corresponds to the y-coordinate and cosine to the x-coordinate. Tangent is calculated as the ratio of sine to cosine (y/x). For example, at 53 degrees, sine is 4/5 and cosine is 3/5, leading to a tangent of 4/3. Understanding these relationships simplifies finding trig values across all quadrants, including negative values in the third quadrant, reinforcing the concept of angles as right triangles within the unit circle.

Sine, Cosine, & Tangent on the Unit Circle

The unit circle is a fundamental concept in trigonometry, where every angle corresponds to a specific point defined by coordinates (x, y) on the circle. The relationship between angles and these coordinates is established through trigonometric functions: sine, cosine, and tangent. These functions can be understood in the context of right triangles formed by the radius of the unit circle, which has a length of 1.

For any angle θ, the sine function is defined as the y-coordinate of the corresponding point on the unit circle, while the cosine function is defined as the x-coordinate. This means that for an angle of 53 degrees, if the coordinates are (3/5, 4/5), then:

$$\sin(53^\circ) = \frac{4}{5}$$

$$\cos(53^\circ) = \frac{3}{5}$$

To remember this relationship, one can note that cosine (c) comes before sine (s) alphabetically, which corresponds to the x-coordinate being the cosine value and the y-coordinate being the sine value.

For the tangent function, it is calculated as the ratio of sine to cosine, or equivalently, the ratio of the y-coordinate to the x-coordinate:

$$\tan(θ) = \frac{\sin(θ)}{\cos(θ)} = \frac{y}{x}$$

For example, at 0 degrees, the coordinates are (1, 0), leading to:

$$\sin(0^\circ) = 0$$

$$\cos(0^\circ) = 1$$

$$\tan(0^\circ) = \frac{0}{1} = 0$$

At 90 degrees, the coordinates are (0, 1), resulting in:

$$\sin(90^\circ) = 1$$

$$\cos(90^\circ) = 0$$

$$\tan(90^\circ) = \frac{1}{0} \text{ (undefined)}$$

Continuing with other angles, at 180 degrees, the coordinates are (-1, 0), giving:

$$\sin(180^\circ) = 0$$

$$\cos(180^\circ) = -1$$

$$\tan(180^\circ) = \frac{0}{-1} = 0$$

At 270 degrees, the coordinates are (0, -1), leading to:

$$\sin(270^\circ) = -1$$

$$\cos(270^\circ) = 0$$

$$\tan(270^\circ) = \frac{-1}{0} \text{ (undefined)}$$

For angles in other quadrants, such as 217 degrees, the coordinates are found in the third quadrant, where both x and y values are negative. For 217 degrees, if the coordinates are (-4/5, -3/5), then:

$$\sin(217^\circ) = -\frac{3}{5}$$

$$\cos(217^\circ) = -\frac{4}{5}$$

$$\tan(217^\circ) = \frac{-\frac{3}{5}}{-\frac{4}{5}} = \frac{3}{4}$$

In the first quadrant, for an angle of 60 degrees, the coordinates are (1/2, √3/2), leading to:

$$\sin(60^\circ) = \frac{\sqrt{3}}{2}$$

$$\cos(60^\circ) = \frac{1}{2}$$

$$\tan(60^\circ) = \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}} = \sqrt{3}$$

Understanding these relationships allows for easier calculations of trigonometric values using the unit circle, reinforcing the connection between angles and their corresponding coordinates.

Find the sine, cosine, and tangent of each angle using the unit circle.
$\theta=-1.18$ rad, $\left(\frac{5}{13},-\frac{12}{13}\right)$

• A. $\sin\theta=-\frac{12}{13},\cos\theta=\frac{5}{13},\tan\theta=\frac{12}{5}$
• B. $\sin\theta=-\frac{12}{13},\cos\theta=\frac{5}{13},\tan\theta=-\frac{12}{5}$
• C. $\sin\theta=\frac{12}{13},\cos\theta=\frac{5}{13},\tan\theta=\frac{5}{12}$
• D. $\sin\theta=\frac{5}{13},\cos\theta=\frac{-12}{13},\tan\theta=\frac{5}{12}$

Answer: (B)

Find the sine, cosine, and tangent of each angle using the unit circle.
$\theta=225\degree,\left(-\frac{\sqrt2}{2},-\frac{\sqrt2}{2}\right)$

• A. $\sin\theta=-\frac{\sqrt2}{2},\cos\theta=-\frac{\sqrt2}{2},\tan\theta=2$
• B. $\sin\theta=\frac{\sqrt2}{2},\cos\theta=-\frac{\sqrt2}{2},\tan\theta=-1$
• C. $\sin\theta=-\frac{\sqrt2}{2},\cos\theta=-\frac{\sqrt2}{2},\tan\theta=1$
• D. $\sin\theta=\frac{\sqrt2}{2},\cos\theta=\frac{\sqrt2}{2},\tan\theta=\frac{\sqrt2}{1}$

Answer: (C)

Here’s what students ask on this topic:

What are the sine and cosine values for common angles on the unit circle?

On the unit circle, common angles and their sine and cosine values are as follows:

• 0°: cos(0°) = 1, sin(0°) = 0
• 30°: cos(30°) = √3/2, sin(30°) = 1/2
• 45°: cos(45°) = √2/2, sin(45°) = √2/2
• 60°: cos(60°) = 1/2, sin(60°) = √3/2
• 90°: cos(90°) = 0, sin(90°) = 1

These values are derived from the coordinates of points on the unit circle corresponding to these angles.

How do you find the tangent of an angle using the unit circle?

To find the tangent of an angle using the unit circle, you divide the sine value by the cosine value of that angle. Mathematically, this is expressed as:

$\frac{\sin }{\cos }$

For example, for 45°, the sine and cosine values are both √2/2. Therefore, the tangent is:

$\frac{\sqrt{2}}{\sqrt{2}}=\; 1$

Why are some trigonometric function values negative on the unit circle?

Trigonometric function values can be negative on the unit circle because the coordinates (x, y) can be negative depending on the quadrant. For example, in the third quadrant, both x and y are negative, making sine and cosine values negative. This reflects the coordinate system where angles are measured, and the signs of the coordinates change based on the quadrant.

How do you use the unit circle to find the sine and cosine of 217 degrees?

To find the sine and cosine of 217 degrees using the unit circle, locate the angle in the third quadrant. The coordinates for 217 degrees are approximately (-4/5, -3/5). Therefore, the sine of 217 degrees is:

$\sin (217°)=-\frac{3}{5}$

And the cosine of 217 degrees is:

$\cos (217°)=-\frac{4}{5}$

What is the relationship between the unit circle and right triangles?

The unit circle and right triangles are related because any angle on the unit circle can be represented as a right triangle. The hypotenuse of this triangle is the radius of the unit circle (which is 1), and the legs are the x and y coordinates of the point on the circle. The sine of the angle is the y-coordinate (opposite side), and the cosine is the x-coordinate (adjacent side).