Trigonometric Functions on the Unit Circle: Sine, Cosine, and Tangent

Trigonometric Functions on the Unit Circle

Sine, Cosine, and Tangent on the Unit Circle

Trigonometric functions relate angles to points on the unit circle. The unit circle is a circle with a radius of 1 centered at the origin (0,0) in the coordinate plane. The coordinates of a point on the unit circle corresponding to an angle θ are (cos θ, sin θ).

• Sine (sin θ): The sine of an angle θ is the y-coordinate of the corresponding point on the unit circle. $\sin \theta = y$

• Cosine (cos θ): The cosine of an angle θ is the x-coordinate of the corresponding point on the unit circle. $\cos \theta = x$

• Tangent (tan θ): The tangent of an angle θ is the ratio of the y-coordinate to the x-coordinate. $\tan \theta = \frac{y}{x}$

Unit Circle Diagram

The unit circle is divided into four quadrants, and key angles (such as 0°, 90°, 180°, 270°, and 360°) correspond to specific points on the circle. The coordinates of these points are used to determine the values of sine, cosine, and tangent for those angles.

• Quadrant I: Both sine and cosine are positive.

• Quadrant II: Sine is positive, cosine is negative.

• Quadrant III: Both sine and cosine are negative.

• Quadrant IV: Sine is negative, cosine is positive.

Examples

Let us find the sine, cosine, and tangent of specific angles using the unit circle:

• Example 1: $\theta = 217^\circ$ The coordinates on the unit circle are approximately (-0.7986, -0.6018).

  • $\sin 217^\circ = -0.6018$

  • $\cos 217^\circ = -0.7986$

  • $\tan 217^\circ = \frac{-0.6018}{-0.7986} \approx 0.7538$

• Example 2: $\theta = 327^\circ$ The coordinates on the unit circle are approximately (0.7986, -0.6018).

  • $\sin 327^\circ = -0.6018$

  • $\cos 327^\circ = 0.7986$

  • $\tan 327^\circ = \frac{-0.6018}{0.7986} \approx -0.7538$

Practice Problems

Use the unit circle to find the sine, cosine, and tangent of the following angles:

• Practice A: $\theta = 121^\circ$

  • $\sin 121^\circ = 0.8572$

  • $\cos 121^\circ = -0.5150$

  • $\tan 121^\circ = \frac{0.8572}{-0.5150} \approx -1.664$

• Practice B: $\theta = 22^\circ$

  • $\sin 22^\circ = 0.3746$

  • $\cos 22^\circ = 0.9272$

  • $\tan 22^\circ = \frac{0.3746}{0.9272} \approx 0.404$

Summary Table: Sine, Cosine, and Tangent on the Unit Circle

Angle (θ)	sin θ	cos θ	tan θ
0°	0	1	0
90°	1	0	undefined
180°	0	-1	0
270°	-1	0	undefined
360°	0	1	0

Additional info: The values for sine, cosine, and tangent at other angles can be found using the coordinates of the corresponding point on the unit circle. The tangent function is undefined when the cosine (x-coordinate) is zero.