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QuestionRewrite sin⁡θcot⁡θ+4cos⁡θ\sin \theta \cot \theta + 4 \cos \theta using sin⁡θ\sin \theta and cos⁡θ\cos \theta, then simplify.

Studdy Solution

STEP 1

Assumptions1. We are given the expression sin⁡θcot⁡θ+4cos⁡θ\sin \theta \cot \theta+4 \cos \theta. . We need to rewrite this expression in terms of sin⁡θ\sin \theta and cos⁡θ\cos \theta.
3. We know that cot⁡θ\cot \theta is the reciprocal of tan⁡θ\tan \theta, which is cos⁡θsin⁡θ\frac{\cos \theta}{\sin \theta}.

STEP 2

Substitute cot⁡θ\cot \theta with cos⁡θsin⁡θ\frac{\cos \theta}{\sin \theta} in the given expression.
sin⁡θcot⁡θ+4cos⁡θ=sin⁡θ(cos⁡θsin⁡θ)+4cos⁡θ\sin \theta \cot \theta+4 \cos \theta = \sin \theta \left(\frac{\cos \theta}{\sin \theta}\right) +4 \cos \theta

STEP 3

implify the expression by cancelling out the common factors.
sin⁡θ(cos⁡θsin⁡θ)+cos⁡θ=cos⁡θ+cos⁡θ\sin \theta \left(\frac{\cos \theta}{\sin \theta}\right) + \cos \theta = \cos \theta + \cos \theta

STEP 4

Combine like terms.
cos⁡θ+4cos⁡θ=cos⁡θ\cos \theta +4 \cos \theta = \cos \thetaSo, the expression sin⁡θcot⁡θ+4cos⁡θ\sin \theta \cot \theta+4 \cos \theta simplifies to cos⁡θ \cos \theta in terms of sin⁡θ\sin \theta and cos⁡θ\cos \theta.

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