Question

Which of the following are true? Select all that apply.
(1 point)
Responses

tangent theta equals start fraction cosine theta over sine theta end fraction
Image with alt text: tangent theta equals start fraction cosine theta over sine theta end fraction

cotangent theta equals start fraction cosine theta over sine theta end fraction
Image with alt text: cotangent theta equals start fraction cosine theta over sine theta end fraction

secant theta equals start fraction 1 over sine theta end fraction
Image with alt text: secant theta equals start fraction 1 over sine theta end fraction

cosecant theta equals start fraction 1 over cosine theta end fraction
Image with alt text: cosecant theta equals start fraction 1 over cosine theta end fraction

secant theta equals start fraction tangent theta over sine theta end fraction
Image with alt text: secant theta equals start fraction tangent theta over sine theta end fraction

Answer 1

To determine which of the statements are true, we need to know the definitions of the trigonometric functions involved:

1. **Tangent (tan)**
\[
\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}
\]

2. **Cotangent (cot)**
\[
\cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)}
\]

3. **Secant (sec)**
\[
\sec(\theta) = \frac{1}{\cos(\theta)}
\]

4. **Cosecant (csc)**
\[
\csc(\theta) = \frac{1}{\sin(\theta)}
\]

Now let's evaluate the statements provided:

- **Tangent theta equals start fraction cosine theta over sine theta end fraction**:
This is **False**. It should be \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \).

- **Cotangent theta equals start fraction cosine theta over sine theta end fraction**:
This is **True**. This is the correct definition of cotangent.

- **Secant theta equals start fraction 1 over sine theta end fraction**:
This is **False**. The correct definition is \( \sec(\theta) = \frac{1}{\cos(\theta)} \).

- **Cosecant theta equals start fraction 1 over cosine theta end fraction**:
This is **False**. The correct definition is \( \csc(\theta) = \frac{1}{\sin(\theta)} \).

- **Secant theta equals start fraction tangent theta over sine theta end fraction**:
This is **False**. To verify, we can use the definitions:
\[
\sec(\theta) = \frac{1}{\cos(\theta)} \quad \text{and} \quad \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}
\]
Thus,
\[
\frac{\tan(\theta)}{\sin(\theta)} = \frac{\frac{\sin(\theta)}{\cos(\theta)}}{\sin(\theta)} = \frac{1}{\cos(\theta)} = \sec(\theta)
\]
However, the given statement does not represent this accurately as it suggests \( \sec(\theta) = \frac{\tan(\theta)}{\sin(\theta)} \) without proper context.

In summary, the only true statement is:

- **cotangent theta equals start fraction cosine theta over sine theta end fraction**.