Question 7 of 10Check all that apply. If sin = , then:A.cos = B.csc = C.tan = D.sec =

To solve the question regarding the trigonometric functions given $\sin$, we will analyze the relationships between the trigonometric ratios involved.

1. Break Down the Problem

We need to determine the values of different trigonometric ratios based on a given value of $\sin$. The options we need to evaluate are:

• A. $\cos$
• B. $\csc$
• C. $\tan$
• D. $\sec$

2. Relevant Concepts

1. Sine Function: $\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}$

2. Cosecant Function: $\csc(\theta) = \frac{1}{\sin(\theta)}$

3. Cosine Function: $\cos(\theta) = \sqrt{1 - \sin^2(\theta)}$

4. Tangent Function: $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$

5. Secant Function: $\sec(\theta) = \frac{1}{\cos(\theta)}$

3. Analysis and Detail

• If $\sin(\theta)$ is known, we can find $\csc(\theta)$ directly since it's the reciprocal of sine.
• To find $\cos(\theta)$, we use the Pythagorean identity $\cos^2(\theta) + \sin^2(\theta) = 1$.
• We can then use $\sin$ and $\cos$ to find $\tan(\theta)$ and $\sec(\theta)$.

4. Verify and Summarize

1. Calculating $\csc$: $\csc(\theta) = \frac{1}{\sin(\theta)}$

2. Calculating $\cos$: Using the identity: $\cos(\theta) = \sqrt{1 - \sin^2(\theta)}$

3. Calculating $\tan$: Once $\cos(\theta)$ is determined: $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$

4. Calculating $\sec$: $\sec(\theta) = \frac{1}{\cos(\theta)}$

Final Answer

The values of the trigonometric functions depend on the given $\sin(\theta)$. Hence, if $\sin = x$ (where $x$ is a specific value), the applicable answers are:

• B. $\csc = \frac{1}{x}$ is valid.
• A. $\cos$ can be calculated as $\sqrt{1 - x^2}$ (valid).
• C. $\tan = \frac{x}{\sqrt{1 - x^2}}$ (valid).
• D. $\sec = \frac{1}{\sqrt{1 - x^2}}$ (valid).

Thus, all options (A, B, C, D) apply based on the relationships derived from $\sin(\theta)$.