Trigonometric Values of Special Angles

Podcast

Play an AI-generated podcast conversation about this lesson

Questions and Answers

Explain why $\sin(0)$ and $\cos(90)$ both evaluate to 0. What does this indicate about their relationship to each other on the unit circle?

Sine corresponds to the y-coordinate and is 0 at 0 degrees. Cosine corresponds to the x-coordinate and is 0 at 90 degrees. This reflects the complementary relationship between sine and cosine, where $\sin(x) = \cos(90 - x)$.

For what angle, $\theta$, are the sine and cosine functions equal? What is the value of $\theta$ in degrees?

$\sin(\theta) = \cos(\theta)$ when $\theta = 45$ degrees.

The value of $\tan(45)$ is 1. Knowing that $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$, explain why $\tan(45)$ evaluates to 1.

At 45 degrees, $\sin(45) = \cos(45)$, so $\frac{\sin(45)}{\cos(45)} = 1$.

What is the relationship between the values of $\tan(30)$ and $\tan(60)$ and why does this relationship exist?

$\tan(30) = \frac{\sqrt{3}}{3}$ and $\tan(60) = \sqrt{3}$. They are reciprocals because 30 and 60 degrees are complementary angles within a right triangle. Signup and view all the answers

Explain why $\csc(30)$ is equal to 2, given that $\sin(30) = \frac{1}{2}$.

Cosecant is the reciprocal of sine. Therefore $\csc(30) = \frac{1}{\sin(30)} = \frac{1}{\frac{1}{2}} = 2$. Signup and view all the answers

Describe how the values of $\sin(\theta)$ change as $\theta$ increases from 0° to 90°.

As $\theta$ increases from 0° to 90°, $\sin(\theta)$ increases from 0 to 1. Signup and view all the answers

Why is $\sec(0)$ equal to 1, and how does this relate to the value of $\cos(0)$?

Since $\sec(\theta)$ is the reciprocal of $\cos(\theta)$, $\sec(0) = \frac{1}{\cos(0)} = \frac{1}{1} = 1$. Signup and view all the answers

Determine the value of $\cot(45)$ and explain how it relates to the value of $\tan(45)$.

$\cot(45) = 1$. Cotangent is the reciprocal of tangent, and since $\tan(45) = 1$, $\cot(45)$ is also 1. Signup and view all the answers

If $\sin(30) = \frac{1}{2}$, what is the value of $\cos(60)$? Explain your reasoning.

$\cos(60) = \frac{1}{2}$. Because sine and cosine are cofunctions and 30 and 60 are complementary angles, $\sin(30) = \cos(90-30) = \cos(60)$. Signup and view all the answers

Flashcards

sin 30°

The sine of 30 degrees is equal to 1/2.

sin 45°

The sine of 45 degrees is equal to √2/2.