Math  /  Trigonometry

QuestionEvaluate without using a calculator. NOTE: If the answer is undefined, indicate that using the check box. (a) cos270=\cos 270^{\circ}= \square 0 Undefined (b) tan270\tan 270^{\circ} Undefined (c) cos540=\cos 540^{\circ}= \square Undefined (d) tan540=\tan 540^{\circ}= \square Undefined (e) tan30=\tan -30^{\circ}= \square Undefined

Studdy Solution

STEP 1

What is this asking? Find the cosine and tangent of some specific angles without a calculator! Watch out! Remember the difference between radians and degrees, and don't mix them up!
Also, recall that some angles have undefined tangent values.

STEP 2

1. Cosine of 270 degrees
2. Tangent of 270 degrees
3. Cosine of 540 degrees
4. Tangent of 540 degrees
5. Tangent of -30 degrees

STEP 3

270270^\circ is located on the negative y-axis of the unit circle.
The cosine represents the x-coordinate.

STEP 4

On the negative y-axis, the x-coordinate is **0**.
So, cos(270)=0\cos(270^\circ) = \boxed{0}.

STEP 5

The tangent of an angle is defined as tan(θ)=sin(θ)cos(θ)\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}.

STEP 6

We know sin(270)=1\sin(270^\circ) = -1 and cos(270)=0\cos(270^\circ) = 0.
Therefore, tan(270)=10\tan(270^\circ) = \frac{-1}{0}.

STEP 7

Dividing by zero is **undefined**.
So, tan(270)\tan(270^\circ) is **undefined**.
Check the **undefined** box.

STEP 8

540540^\circ is more than a full rotation around the unit circle.
Let's find an equivalent angle within one rotation.

STEP 9

Subtracting a full rotation (360360^\circ) from 540540^\circ gives us 540360=180540^\circ - 360^\circ = 180^\circ.

STEP 10

180180^\circ is on the negative x-axis.
The cosine represents the x-coordinate, which is **-1**.

STEP 11

Therefore, cos(540)=cos(180)=1\cos(540^\circ) = \cos(180^\circ) = \boxed{-1}.

STEP 12

Since 540540^\circ is equivalent to 180180^\circ, we have tan(540)=tan(180)\tan(540^\circ) = \tan(180^\circ).

STEP 13

We know sin(180)=0\sin(180^\circ) = 0 and cos(180)=1\cos(180^\circ) = -1.

STEP 14

So, tan(180)=sin(180)cos(180)=01=0\tan(180^\circ) = \frac{\sin(180^\circ)}{\cos(180^\circ)} = \frac{0}{-1} = 0.

STEP 15

Therefore, tan(540)=0\tan(540^\circ) = \boxed{0}.

STEP 16

A negative angle means we rotate clockwise. 30-30^\circ is the same as 36030=330360^\circ - 30^\circ = 330^\circ.

STEP 17

We can use the fact that tan(30)=tan(30)\tan(-30^\circ) = -\tan(30^\circ).

STEP 18

We know that tan(30)=sin(30)cos(30)=1232=13=33\tan(30^\circ) = \frac{\sin(30^\circ)}{\cos(30^\circ)} = \frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}} = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}.

STEP 19

Therefore, tan(30)=tan(30)=33\tan(-30^\circ) = -\tan(30^\circ) = \boxed{-\frac{\sqrt{3}}{3}}.

STEP 20

(a) cos270=0\cos 270^{\circ}= 0 (b) tan270\tan 270^{\circ} is Undefined (c) cos540=1\cos 540^{\circ}= -1 (d) tan540=0\tan 540^{\circ}= 0 (e) tan30=33\tan -30^{\circ}= -\frac{\sqrt{3}}{3}

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