Determine $\cos \theta$ When $\sin (-\theta)=\frac{9}{13}$ And \$\tan \theta=\frac{5 \sqrt{5}}{7}$[/tex\]. Leave Your Answer As A Fraction.

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**Determine $\cos \theta$ when $\sin (-\theta)=\frac{9}{13}$ and \$\tan \theta=\frac{5 \sqrt{5}}{7}$**

Introduction In trigonometry, we often encounter problems that involve finding the values of trigonometric functions given certain conditions. In this article, we will explore how to determine the value of $\cos \theta$ when given the values of $\sin (-\theta)$ and $\tan \theta$.

Understanding the Given Information We are given two pieces of information:

  • sin⁡(−θ)=913\sin (-\theta)=\frac{9}{13}

  • tan⁡θ=557\tan \theta=\frac{5 \sqrt{5}}{7}

Recalling Trigonometric Identities To solve this problem, we need to recall some trigonometric identities. Specifically, we need to remember the following:

  • sin⁡(−θ)=−sin⁡θ\sin (-\theta) = -\sin \theta

  • tan⁡θ=sin⁡θcos⁡θ\tan \theta = \frac{\sin \theta}{\cos \theta}

Using the Given Information to Find $\sin \theta$ We are given that $\sin (-\theta)=\frac{9}{13}$. Using the identity $\sin (-\theta) = -\sin \theta$, we can rewrite this as:

−sin⁡θ=913-\sin \theta = \frac{9}{13}

Multiplying both sides by $-1$, we get:

sin⁡θ=−913\sin \theta = -\frac{9}{13}

Using the Given Information to Find $\cos \theta$ We are also given that $\tan \theta=\frac{5 \sqrt{5}}{7}$. Using the identity $\tan \theta = \frac{\sin \theta}{\cos \theta}$, we can rewrite this as:

sin⁡θcos⁡θ=557\frac{\sin \theta}{\cos \theta} = \frac{5 \sqrt{5}}{7}

Substituting the value of $\sin \theta$ we found earlier, we get:

−913cos⁡θ=557\frac{-\frac{9}{13}}{\cos \theta} = \frac{5 \sqrt{5}}{7}

Multiplying both sides by $\cos \theta$, we get:

−913=557cos⁡θ-\frac{9}{13} = \frac{5 \sqrt{5}}{7} \cos \theta

Multiplying both sides by $\frac{13}{5 \sqrt{5}}$, we get:

−913⋅1355=cos⁡θ-\frac{9}{13} \cdot \frac{13}{5 \sqrt{5}} = \cos \theta

Simplifying, we get:

cos⁡θ=−955\cos \theta = -\frac{9}{5 \sqrt{5}}

Rationalizing the denominator, we get:

cos⁡θ=−9525\cos \theta = -\frac{9 \sqrt{5}}{25}

Conclusion In this article, we used the given information to find the value of $\cos \theta$ when $\sin (-\theta)=\frac{9}{13}$ and $\tan \theta=\frac{5 \sqrt{5}}{7}$. We used trigonometric identities to rewrite the given information and then solved for $\cos \theta$.

Q&A

**Q: What is the value of $\cos \theta$ when $\sin (-\theta)=\frac9}{13}$ and $\tan \theta=\frac{5 \sqrt{5}}{7}$?** A $\cos \theta = -\frac{9 \sqrt{5}{25}$

Q: How do I use trigonometric identities to rewrite the given information? A: You can use the identities $\sin (-\theta) = -\sin \theta$ and $\tan \theta = \frac{\sin \theta}{\cos \theta}$ to rewrite the given information.

Q: What is the significance of rationalizing the denominator? A: Rationalizing the denominator is important because it allows us to simplify the expression and make it easier to work with.

Q: Can I use this method to find the value of $\cos \theta$ when given different values of $\sin (-\theta)$ and $\tan \theta$? A: Yes, you can use this method to find the value of $\cos \theta$ when given different values of $\sin (-\theta)$ and $\tan \theta$. Just remember to use the trigonometric identities to rewrite the given information and then solve for $\cos \theta$.