Studdy AI Logo
Math  /  Trigonometry

Questiond. csc⁡(tan⁡−1(−2))\csc \left(\tan ^{-1}(-2)\right)

Studdy Solution

STEP 1

1. We need to find the value of the cosecant function given an inverse tangent function.
2. The inverse tangent function, tan⁡−1(−2)\tan^{-1}(-2), gives an angle whose tangent is −2-2.
3. The cosecant function is the reciprocal of the sine function.

STEP 2

1. Determine the angle θ\theta such that tan⁡(θ)=−2\tan(\theta) = -2.
2. Use the identity sin⁡(θ)=11+tan⁡2(θ)\sin(\theta) = \frac{1}{\sqrt{1+\tan^2(\theta)}} to find sin⁡(θ)\sin(\theta).
3. Calculate csc⁡(θ)\csc(\theta) as the reciprocal of sin⁡(θ)\sin(\theta).

STEP 3

Let θ=tan⁡−1(−2)\theta = \tan^{-1}(-2). This implies that tan⁡(θ)=−2\tan(\theta) = -2.

STEP 4

Use the identity for sine in terms of tangent:
sin⁡(θ)=tan⁡(θ)1+tan⁡2(θ) \sin(\theta) = \frac{\tan(\theta)}{\sqrt{1+\tan^2(\theta)}}
Substitute tan⁡(θ)=−2\tan(\theta) = -2:
sin⁡(θ)=−21+(−2)2 \sin(\theta) = \frac{-2}{\sqrt{1+(-2)^2}}
sin⁡(θ)=−21+4 \sin(\theta) = \frac{-2}{\sqrt{1+4}}
sin⁡(θ)=−25 \sin(\theta) = \frac{-2}{\sqrt{5}}

STEP 5

Calculate csc⁡(θ)\csc(\theta), which is the reciprocal of sin⁡(θ)\sin(\theta):
csc⁡(θ)=1sin⁡(θ)=5−2 \csc(\theta) = \frac{1}{\sin(\theta)} = \frac{\sqrt{5}}{-2}
csc⁡(θ)=−52 \csc(\theta) = -\frac{\sqrt{5}}{2}
The value of csc⁡(tan⁡−1(−2))\csc \left(\tan ^{-1}(-2)\right) is:
−52 \boxed{-\frac{\sqrt{5}}{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