Studdy AI Logo
Math  /  Trigonometry

QuestionUse a compound angle formula to determine a trigonometric expression that is equivalent to sin⁡(x−4π3)\sin \left(x-\frac{4 \pi}{3}\right).

Studdy Solution

STEP 1

1. We are using the compound angle formula for sine.
2. The compound angle formula for sine is sin⁡(a−b)=sin⁡acos⁡b−cos⁡asin⁡b\sin(a - b) = \sin a \cos b - \cos a \sin b.
3. We will apply this formula to the expression sin⁡(x−4π3)\sin \left(x-\frac{4 \pi}{3}\right).

STEP 2

1. Identify the values of aa and bb.
2. Apply the compound angle formula.
3. Simplify the resulting expression using known trigonometric values.

STEP 3

Identify the values of aa and bb in the expression sin⁡(x−4π3)\sin \left(x-\frac{4 \pi}{3}\right).
Here, a=xa = x and b=4π3b = \frac{4\pi}{3}.

STEP 4

Apply the compound angle formula sin⁡(a−b)=sin⁡acos⁡b−cos⁡asin⁡b\sin(a - b) = \sin a \cos b - \cos a \sin b to the expression sin⁡(x−4π3)\sin \left(x-\frac{4 \pi}{3}\right).
sin⁡(x−4π3)=sin⁡xcos⁡(4π3)−cos⁡xsin⁡(4π3)\sin \left(x-\frac{4 \pi}{3}\right) = \sin x \cos \left(\frac{4 \pi}{3}\right) - \cos x \sin \left(\frac{4 \pi}{3}\right)

STEP 5

Simplify the expression using known trigonometric values for 4π3\frac{4\pi}{3}.
The angle 4π3\frac{4\pi}{3} is in the third quadrant where sine is negative and cosine is negative. The reference angle is π3\frac{\pi}{3}.
cos⁡(4π3)=−cos⁡(π3)=−12\cos \left(\frac{4 \pi}{3}\right) = -\cos \left(\frac{\pi}{3}\right) = -\frac{1}{2}
sin⁡(4π3)=−sin⁡(π3)=−32\sin \left(\frac{4 \pi}{3}\right) = -\sin \left(\frac{\pi}{3}\right) = -\frac{\sqrt{3}}{2}
Substitute these values back into the expression:
sin⁡(x−4π3)=sin⁡x(−12)−cos⁡x(−32)\sin \left(x-\frac{4 \pi}{3}\right) = \sin x \left(-\frac{1}{2}\right) - \cos x \left(-\frac{\sqrt{3}}{2}\right)
Simplify:
sin⁡(x−4π3)=−12sin⁡x+32cos⁡x\sin \left(x-\frac{4 \pi}{3}\right) = -\frac{1}{2} \sin x + \frac{\sqrt{3}}{2} \cos x
The trigonometric expression equivalent to sin⁡(x−4π3)\sin \left(x-\frac{4 \pi}{3}\right) is −12sin⁡x+32cos⁡x-\frac{1}{2} \sin x + \frac{\sqrt{3}}{2} \cos x.

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