Decide whether the following statement is true or false.
If theta is an acute angle and secant thetaequals22 then cosine theta equals StartFraction 1 Over 22 EndFraction
.
Question content area bottom
Part 1
Choose the correct answer below.
A.
​False, because secant theta equals one half cosine theta
. ​Thus, if secant thetaequals22​, then cosine theta equals 44.
B.
​False, because secant theta equals cosine theta. ​Thus, if secant thetaequals22​, then cosine theta equals 22.
C.
​True, because secant theta equals StartFraction 1 Over cosine theta EndFraction
. That means cosine theta equals StartFraction 1 Over secant theta EndFraction
. ​Thus, if secant thetaequals22​, then cosine theta equals StartFraction 1 Over 22 EndFraction
.
D.
​True, because secant theta equals cosine left parenthesis StartFraction 1 Over theta EndFraction right parenthesis
. ​Thus, if secant thetaequals22​, then cosine theta equals StartFraction 1 Over 22 EndFraction
.

The correct answer is:

**C. True, because secant θ equals \( \frac{1}{\cos(\theta)} \). That means \(\cos(\theta) = \frac{1}{\sec(\theta)}\). Thus, if \(\sec(\theta) = 22\), then \(\cos(\theta) = \frac{1}{22}\).**

### Explanation:
- The secant function is defined as the reciprocal of the cosine function:
  \[
  \sec(\theta) = \frac{1}{\cos(\theta)}
  \]
- Therefore, if \(\sec(\theta) = 22\), then
  \[
  \cos(\theta) = \frac{1}{\sec(\theta)} = \frac{1}{22}
  \]
  
This reasoning matches option **C**. Options **A**, **B**, and **D** contain incorrect interpretations of the secant and cosine functions.

Would you like more details or have any questions about this?

Here are 5 related questions for further practice:
1. What is the cosine of an angle if the secant of that angle is 5?
2. If \(\cos(\theta) = \frac{1}{10}\), what is \(\sec(\theta)\)?
3. How is secant related to sine and tangent in trigonometric identities?
4. What is the range of values that \(\sec(\theta)\) can take for an acute angle \(\theta\)?
5. If \(\sec(\theta) = -4\), can \(\theta\) be an acute angle?

**Tip:** Remember that secant and cosine are reciprocals, so \(\sec(\theta)\) will be large when \(\cos(\theta)\) is small, and vice versa.

Decide whether the following statement is true or false.
If theta is an acute angle and secant theta equals 22, then cosine theta equals 1/22.
Choose the correct answer below.
A. False, because secant theta equals one half cosine theta. Thus, if secant theta equals 22, then cosine theta equals 44.
B. False, because secant theta equals cosine theta. Thus, if secant theta equals 22, then cosine theta equals 22.
C. True, because secant theta equals 1/cosine theta. That means cosine theta equals 1/secant theta. Thus, if secant theta equals 22, then cosine theta equals 1/22.
D. True, because secant theta equals cosine (1/theta). Thus, if secant theta equals 22, then cosine theta equals 1/22.

Learn how to determine cosine theta when given secant theta for an acute angle. This trigonometry problem explores the relationship between secant and cosine, showing how cosine theta equals 1/22 when secant theta equals 22. Suitable for high school students studying trigonometric functions and identities.

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