Questions: Use the definition or identities to find the exact value of each of the remaining five trigonometric functions of the acute angle θ. tan θ = √29 sin θ = □ (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression. Type an exact answer, using radicals as needed.)

Solution Steps

To find the exact value of the remaining five trigonometric functions given $\tan \theta = \sqrt{29}$, we can use the following steps:

1. Use the definition of $\tan \theta$ to find $\sin \theta$ and $\cos \theta$.
2. Use the Pythagorean identity to find $\sin \theta$ and $\cos \theta$.
3. Calculate $\csc \theta$, $\sec \theta$, and $\cot \theta$ using the reciprocal identities.

1. Given $\tan \theta = \sqrt{29}$, we know $\tan \theta = \frac{\sin \theta}{\cos \theta}$.
2. Assume $\sin \theta = \sqrt{29} \cos \theta$.
3. Use the Pythagorean identity: $\sin^2 \theta + \cos^2 \theta = 1$.
4. Solve for $\cos \theta$ and then find $\sin \theta$.
5. Use the reciprocal identities to find $\csc \theta$, $\sec \theta$, and $\cot \theta$.

We are given that $\tan \theta = \sqrt{29}$.

Using the definition of tangent, we have: $$\tan \theta = \frac{\sin \theta}{\cos \theta} \implies \sin \theta = \sqrt{29} \cos \theta$$

Using the Pythagorean identity: $$\sin^2 \theta + \cos^2 \theta = 1$$ Substitute $\sin \theta = \sqrt{29} \cos \theta$: $$(\sqrt{29} \cos \theta)^2 + \cos \theta^2 = 1 \implies 29 \cos^2 \theta + \cos^2 \theta = 1 \implies 30 \cos^2 \theta = 1 \implies \cos^2 \theta = \frac{1}{30}$$ Thus: $$\cos \theta = \frac{\sqrt{30}}{30}$$

Using $\sin \theta = \sqrt{29} \cos \theta$: $$\sin \theta = \sqrt{29} \cdot \frac{\sqrt{30}}{30} = \frac{\sqrt{870}}{30}$$

Using the reciprocal identities: $$\csc \theta = \frac{1}{\sin \theta} = \frac{30}{\sqrt{870}} = \frac{\sqrt{870}}{29}$$ $$\sec \theta = \frac{1}{\cos \theta} = \frac{30}{\sqrt{30}} = \sqrt{30}$$ $$\cot \theta = \frac{1}{\tan \theta} = \frac{1}{\sqrt{29}} = \frac{\sqrt{29}}{29}$$

Final Answer