Math / Trigonometry

QuestionList the value of all three trigonometric functions for the given angle $\pi / 3$ .

Studdy Solution

STEP 1

What is this asking? What are the sine, cosine, and tangent of $\pi / 3$ radians, which is the same as 60 degrees? Watch out! Don't mix up radians and degrees!
Also, remember your special triangles!

STEP 2

1. Draw a triangle
2. Calculate sine
3. Calculate cosine
4. Calculate tangent

STEP 3

Let's visualize this angle using a 30-60-90 triangle!
Remember, $\pi / 3$ radians is the same as 60 degrees.
We'll draw a 30-60-90 triangle, where the 60-degree angle is one of the acute angles.

STEP 4

Recall that in a 30-60-90 triangle, the sides are in a special ratio: $1 : \sqrt{3} : 2$ , where 1 is the length of the side opposite the 30-degree angle, $\sqrt{3}$ is the length of the side opposite the 60-degree angle, and 2 is the length of the hypotenuse.

STEP 5

Remember, sine is defined as the ratio of the opposite side to the hypotenuse.
In our triangle, the side opposite the $\pi / 3$ (60-degree) angle has length $\sqrt{3}$ , and the hypotenuse has length 2.

STEP 6

So, $\sin(\pi / 3) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{\sqrt{3}}{2}.$

STEP 7

Cosine is the ratio of the adjacent side to the hypotenuse.
The side adjacent to the $\pi / 3$ (60-degree) angle has length 1, and the hypotenuse has length 2.

STEP 8

Therefore, $\cos(\pi / 3) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{1}{2}.$

STEP 9

Tangent is the ratio of the opposite side to the adjacent side.
The side opposite the $\pi / 3$ (60-degree) angle has length $\sqrt{3}$ , and the side adjacent to it has length 1.

STEP 10

Thus, $\tan(\pi / 3) = \frac{\text{opposite}}{\text{adjacent}} = \frac{\sqrt{3}}{1} = \sqrt{3}.$

STEP 11

$\sin(\pi / 3) = \frac{\sqrt{3}}{2}$ $\cos(\pi / 3) = \frac{1}{2}$ $\tan(\pi / 3) = \sqrt{3}$

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