Find cscθ, cotθ, and sinθ, where θ is the angle shown in the figure.Give exact values, not decimal approximations.θ78

1. Break Down the Problem

To find $\csc \theta$, $\cot \theta$, and $\sin \theta$ where $\theta = 78^\circ$, we need to use the relationships between these trigonometric functions.

2. Relevant Concepts

The trigonometric identities we will use are:

• $\csc \theta = \frac{1}{\sin \theta}$
• $\cot \theta = \frac{\cos \theta}{\sin \theta}$
• $\sin \theta$ and $\cos \theta$ can be derived from the right triangle formed by $\theta$.

3. Analysis and Detail

1. Finding $\sin \theta$: $\sin 78^\circ$ We can use known values from trigonometric tables or equalities: $\sin 78^\circ = \sin(90^\circ - 12^\circ) = \cos 12^\circ$ The exact value might not be directly available in the common trigonometric identities.

2. Finding $\csc \theta$: $\csc \theta = \frac{1}{\sin 78^\circ}$

3. Finding $\cot \theta$: $\cot \theta = \frac{\cos 78^\circ}{\sin 78^\circ}$ Again, we can express $\cos 78^\circ = \sin(90^\circ - 78^\circ) = \sin 12^\circ$.

4. Verify and Summarize

Now, summarizing the findings:

• $\sin 78^\circ$ is known from trigonometric values.
• Calculate $\csc 78^\circ$ and $\cot 78^\circ$ using the values.

Final Answer

Given that specific exact trigonometric values for $\sin 12^\circ$ and $\cos 12^\circ$ aren’t typically involved in elementary trigonometric identities, we conclude:

• $\sin 78^\circ = \sin 78^\circ$
• $\csc 78^\circ = \frac{1}{\sin 78^\circ}$
• $\cot 78^\circ = \frac{\cos 78^\circ}{\sin 78^\circ}$

Thus, empirical tables or calculators are often necessary to obtain exact values, as calculations from sine and cosine are often required.