Finding Trigonometric Function Values from a Point on the Coordinate Plane

Q10. Find the values of the six trigonometric functions for the angle shown (a.)

Background

Topic: Trigonometric Functions from a Point

This question tests your ability to find the six trigonometric function values (sine, cosine, tangent, cosecant, secant, cotangent) for an angle θ, given a point on the terminal side of θ in the coordinate plane.

Key Terms and Formulas

• Given a point $(x, y)$ on the terminal side of angle θ, the distance from the origin (r) is found by $r = \sqrt{x^2 + y^2}$.

• $\sin \theta = \frac{y}{r}$

• $\cos \theta = \frac{x}{r}$

• $\tan \theta = \frac{y}{x}$

• $\csc \theta = \frac{r}{y}$

• $\sec \theta = \frac{r}{x}$

• $\cot \theta = \frac{x}{y}$

Step-by-Step Guidance

1. Identify the coordinates given: $x = \sqrt{7}$, $y = -3$.

2. Calculate $r$ using the formula $r = \sqrt{x^2 + y^2}$.

3. Plug the values of $x$, $y$, and $r$ into the formulas for $\sin \theta$, $\cos \theta$, and $\tan \theta$.

4. Write the reciprocal functions: $\csc \theta$, $\sec \theta$, and $\cot \theta$ using the values found above.

Try solving on your own before revealing the answer!

Final Answer:

$r = \sqrt{(\sqrt{7})^2 + (-3)^2} = \sqrt{7 + 9} = \sqrt{16} = 4$

$\sin \theta = \frac{-3}{4}$

$\cos \theta = \frac{\sqrt{7}}{4}$

$\tan \theta = \frac{-3}{\sqrt{7}}$

$\csc \theta = \frac{4}{-3}$

$\sec \theta = \frac{4}{\sqrt{7}}$

$\cot \theta = \frac{\sqrt{7}}{-3}$

Each function value is found by substituting the coordinates and $r$ into the respective formulas.