Evaluating Circular Function Values Using the Unit Circle

Q1. The figure shows an angle θ in standard position with its terminal side intersecting the unit circle. Evaluate the six circular function values of θ.

Background

Topic: Trigonometric Functions on the Unit Circle

This question tests your understanding of how to find the values of the six trigonometric functions (sine, cosine, tangent, cosecant, secant, and cotangent) for an angle θ using the unit circle. The unit circle is a fundamental concept in trigonometry, where the coordinates of a point on the circle correspond to the cosine and sine of the angle.

Key Terms and Formulas

• Unit Circle: A circle with radius 1 centered at the origin (0,0).

• Coordinates: If the terminal side of θ intersects the unit circle at $(x, y)$, then:

$\sin \theta = y$

$\cos \theta = x$

$\tan \theta = \frac{y}{x}$

$\csc \theta = \frac{1}{y}$

$\sec \theta = \frac{1}{x}$

$\cot \theta = \frac{x}{y}$

Step-by-Step Guidance

1. Identify the coordinates where the terminal side of θ intersects the unit circle. According to the diagram, these are $\left(\frac{5}{13}, \frac{12}{13}\right)$.

2. Recall that for any angle θ in standard position, the x-coordinate gives $\cos \theta$ and the y-coordinate gives $\sin \theta$.

3. Write the values for $\sin \theta$ and $\cos \theta$ using the coordinates:

   • $\sin \theta = \frac{12}{13}$

   • $\cos \theta = \frac{5}{13}$

4. Set up the expressions for the other four trigonometric functions using the values above:

   • $\tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{12/13}{5/13}$

   • $\csc \theta = \frac{1}{\sin \theta} = \frac{1}{12/13}$

   • $\sec \theta = \frac{1}{\cos \theta} = \frac{1}{5/13}$

   • $\cot \theta = \frac{\cos \theta}{\sin \theta} = \frac{5/13}{12/13}$

Try solving on your own before revealing the answer!

Final Answer:

• $\sin \theta = \frac{12}{13}$

• $\cos \theta = \frac{5}{13}$

• $\tan \theta = \frac{12}{5}$

• $\csc \theta = \frac{13}{12}$

• $\sec \theta = \frac{13}{5}$

• $\cot \theta = \frac{5}{12}$

Each function is evaluated using the coordinates from the unit circle. The denominators and numerators simplify because the radius is 1, so the coordinates are already normalized.