Studdy AI Logo
Math  /  Trigonometry

Question23. Evaluate the functions. a) sin⁡4π3\sin \frac{4 \pi}{3} b) tan⁡(−225)\tan (-225) c) sec⁡11π6\sec \frac{11 \pi}{6} d) sec⁡9π2\sec \frac{9 \pi}{2} e) sin⁡(−510)\sin (-510)

Studdy Solution

STEP 1

1. We are evaluating trigonometric functions for given angles.
2. Angles may need to be converted to a standard position within the unit circle.
3. Use the properties of trigonometric functions and their periodicity.

STEP 2

1. Evaluate sin⁡4π3\sin \frac{4 \pi}{3}.
2. Evaluate tan⁡(−225∘)\tan (-225^\circ).
3. Evaluate sec⁡11π6\sec \frac{11 \pi}{6}.
4. Evaluate sec⁡9π2\sec \frac{9 \pi}{2}.
5. Evaluate sin⁡(−510∘)\sin (-510^\circ).

STEP 3

Convert 4π3\frac{4 \pi}{3} to degrees if necessary and determine its reference angle:
4π3×180∘π=240∘\frac{4 \pi}{3} \times \frac{180^\circ}{\pi} = 240^\circ
The reference angle for 240∘240^\circ is 240∘−180∘=60∘240^\circ - 180^\circ = 60^\circ.

STEP 4

Determine the sign of sin⁡\sin in the third quadrant:
sin⁡\sin is negative in the third quadrant.

STEP 5

Evaluate sin⁡240∘\sin 240^\circ:
sin⁡240∘=−sin⁡60∘=−32\sin 240^\circ = -\sin 60^\circ = -\frac{\sqrt{3}}{2}

STEP 6

Convert −225∘-225^\circ to a positive angle:
−225∘+360∘=135∘-225^\circ + 360^\circ = 135^\circ

STEP 7

Determine the reference angle and the sign of tan⁡\tan in the second quadrant:
The reference angle for 135∘135^\circ is 180∘−135∘=45∘180^\circ - 135^\circ = 45^\circ.
tan⁡\tan is negative in the second quadrant.

STEP 8

Evaluate tan⁡135∘\tan 135^\circ:
tan⁡135∘=−tan⁡45∘=−1\tan 135^\circ = -\tan 45^\circ = -1

STEP 9

Convert 11π6\frac{11 \pi}{6} to degrees if necessary and determine its reference angle:
11π6×180∘π=330∘\frac{11 \pi}{6} \times \frac{180^\circ}{\pi} = 330^\circ
The reference angle for 330∘330^\circ is 360∘−330∘=30∘360^\circ - 330^\circ = 30^\circ.

STEP 10

Determine the sign of sec⁡\sec in the fourth quadrant:
sec⁡\sec is positive in the fourth quadrant.

STEP 11

Evaluate sec⁡330∘\sec 330^\circ:
sec⁡330∘=1cos⁡330∘=1cos⁡30∘=132=23=233\sec 330^\circ = \frac{1}{\cos 330^\circ} = \frac{1}{\cos 30^\circ} = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}} = \frac{2\sqrt{3}}{3}

STEP 12

Convert 9π2\frac{9 \pi}{2} to degrees if necessary and determine its equivalent angle:
9π2×180∘π=810∘\frac{9 \pi}{2} \times \frac{180^\circ}{\pi} = 810^\circ
Subtract 360∘360^\circ repeatedly to find an equivalent angle within 0∘0^\circ to 360∘360^\circ:
810∘−720∘=90∘810^\circ - 720^\circ = 90^\circ

STEP 13

Evaluate sec⁡90∘\sec 90^\circ:
sec⁡90∘\sec 90^\circ is undefined because cos⁡90∘=0\cos 90^\circ = 0.

STEP 14

Convert −510∘-510^\circ to a positive angle:
−510∘+720∘=210∘-510^\circ + 720^\circ = 210^\circ

STEP 15

Determine the reference angle and the sign of sin⁡\sin in the third quadrant:
The reference angle for 210∘210^\circ is 210∘−180∘=30∘210^\circ - 180^\circ = 30^\circ.
sin⁡\sin is negative in the third quadrant.

STEP 16

Evaluate sin⁡210∘\sin 210^\circ:
sin⁡210∘=−sin⁡30∘=−12\sin 210^\circ = -\sin 30^\circ = -\frac{1}{2}
The evaluated values are: a) sin⁡4π3=−32\sin \frac{4 \pi}{3} = -\frac{\sqrt{3}}{2} b) tan⁡(−225∘)=−1\tan (-225^\circ) = -1 c) sec⁡11π6=233\sec \frac{11 \pi}{6} = \frac{2\sqrt{3}}{3} d) sec⁡9π2=undefined\sec \frac{9 \pi}{2} = \text{undefined} e) sin⁡(−510∘)=−12\sin (-510^\circ) = -\frac{1}{2}

Was this helpful?

Studdy solves anything!

banner

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

ParentsInfluencer programContactPolicyTerms
TwitterInstagramFacebookTikTokDiscord