Suppose that $\theta$ is an angle in standard position whose terminal side intersects the unit circle at $\left(-\frac{2}{5},-\frac{\sqrt{21}}{5}\right)$.
Find the exact values of $\cos \theta, \csc \theta$, and $\tan \theta$.
$\cos \theta=$
$\square$
$\csc \theta=$
$\square$
$\tan \theta=\square$
Answer (1)
Determine cos θ directly from the x-coordinate of the point on the unit circle: cos θ = − 5 2 .
Calculate csc θ as the reciprocal of the y-coordinate ( sin θ ) and rationalize the denominator: csc θ = − 21 5 21 .
Compute tan θ by dividing the y-coordinate ( sin θ ) by the x-coordinate ( cos θ ): tan θ = 2 21 .
State the final answers: cos θ = − 5 2 , csc θ = − 21 5 21 , tan θ = 2 21 .
Explanation
Analyze the problem and available data We are given that the terminal side of angle θ intersects the unit circle at the point ( − 5 2 , − 5 21 ) . Our goal is to find the exact values of cos θ , csc θ , and tan θ .
Relate coordinates to trigonometric functions Recall that for a point ( x , y ) on the unit circle corresponding to an angle θ in standard position, we have cos θ = x and sin θ = y . In this case, x = − 5 2 and y = − 5 21 .
Find cos θ Therefore, cos θ = x = − 5 2 .
Find csc θ Next, we find csc θ . Recall that csc θ = s i n θ 1 . Since sin θ = y = − 5 21 , we have csc θ = − 5 21 1 = − 21 5 To rationalize the denominator, we multiply the numerator and denominator by 21 :
csc θ = − 21 5 21 .
Find tan θ Finally, we find tan θ . Recall that tan θ = c o s θ s i n θ . We have sin θ = − 5 21 and cos θ = − 5 2 , so tan θ = − 5 2 − 5 21 = 2 21 .
State the final answer Therefore, the exact values are: cos θ = − 5 2 csc θ = − 21 5 21 tan θ = 2 21
Examples
Understanding trigonometric functions on the unit circle is crucial in fields like physics and engineering. For example, when analyzing simple harmonic motion, such as the motion of a pendulum or a spring, the position of the object can be described using trigonometric functions. The cosine and sine of an angle determine the object's coordinates at any given time, allowing engineers to predict and control the system's behavior. This principle extends to more complex systems, such as analyzing alternating current (AC) circuits, where voltage and current vary sinusoidally with time.
