Question
Which of the following are trigonometric identities? Select all that apply.
(2 points)
Responses
tangent theta equals Start Fraction sine theta over cosine theta End Fraction
Image with alt text: tangent theta equals Start Fraction sine theta over cosine theta End Fraction
cosine theta equals Start Fraction 1 over secant theta EnD Fraction
Image with alt text: cosine theta equals Start Fraction 1 over secant theta EnD Fraction
secant theta equals Start Fraction 1 over cosecant theta End Fraction
Image with alt text: secant theta equals Start Fraction 1 over cosecant theta End Fraction
sine squared theta minus cosine squared theta equals 1
Image with alt text: sine squared theta minus cosine squared theta equals 1
1 plus tangent squared theta equals secant squared theta
(2 points)
Responses
tangent theta equals Start Fraction sine theta over cosine theta End Fraction
Image with alt text: tangent theta equals Start Fraction sine theta over cosine theta End Fraction
cosine theta equals Start Fraction 1 over secant theta EnD Fraction
Image with alt text: cosine theta equals Start Fraction 1 over secant theta EnD Fraction
secant theta equals Start Fraction 1 over cosecant theta End Fraction
Image with alt text: secant theta equals Start Fraction 1 over cosecant theta End Fraction
sine squared theta minus cosine squared theta equals 1
Image with alt text: sine squared theta minus cosine squared theta equals 1
1 plus tangent squared theta equals secant squared theta
Answers
GPT-4o mini
To determine which of the given responses are trigonometric identities, let's evaluate each statement:
1. **tangent theta equals Start Fraction sine theta over cosine theta End Fraction**
This statement is true. It states that \(\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\).
**This is a valid trigonometric identity.**
2. **cosine theta equals Start Fraction 1 over secant theta End Fraction**
This statement is true. It states that \(\cos(\theta) = \frac{1}{\sec(\theta)}\).
**This is a valid trigonometric identity.**
3. **secant theta equals Start Fraction 1 over cosecant theta End Fraction**
This statement is false. The correct relationship is \(\sec(\theta) = \frac{1}{\cos(\theta)}\) and \(\csc(\theta) = \frac{1}{\sin(\theta)}\). Therefore, \(\sec(\theta)\) cannot equal \(\frac{1}{\csc(\theta)}\).
**This is not a trigonometric identity.**
4. **sine squared theta minus cosine squared theta equals 1**
This statement is false. The correct identity is \(\sin^2(\theta) + \cos^2(\theta) = 1\).
**This is not a trigonometric identity.**
5. **1 plus tangent squared theta equals secant squared theta**
This statement is true. It states that \(1 + \tan^2(\theta) = \sec^2(\theta)\).
**This is a valid trigonometric identity.**
### Summary:
The valid trigonometric identities from the list provided are:
1. **tangent theta equals Start Fraction sine theta over cosine theta End Fraction**
2. **cosine theta equals Start Fraction 1 over secant theta End Fraction**
5. **1 plus tangent squared theta equals secant squared theta**
1. **tangent theta equals Start Fraction sine theta over cosine theta End Fraction**
This statement is true. It states that \(\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\).
**This is a valid trigonometric identity.**
2. **cosine theta equals Start Fraction 1 over secant theta End Fraction**
This statement is true. It states that \(\cos(\theta) = \frac{1}{\sec(\theta)}\).
**This is a valid trigonometric identity.**
3. **secant theta equals Start Fraction 1 over cosecant theta End Fraction**
This statement is false. The correct relationship is \(\sec(\theta) = \frac{1}{\cos(\theta)}\) and \(\csc(\theta) = \frac{1}{\sin(\theta)}\). Therefore, \(\sec(\theta)\) cannot equal \(\frac{1}{\csc(\theta)}\).
**This is not a trigonometric identity.**
4. **sine squared theta minus cosine squared theta equals 1**
This statement is false. The correct identity is \(\sin^2(\theta) + \cos^2(\theta) = 1\).
**This is not a trigonometric identity.**
5. **1 plus tangent squared theta equals secant squared theta**
This statement is true. It states that \(1 + \tan^2(\theta) = \sec^2(\theta)\).
**This is a valid trigonometric identity.**
### Summary:
The valid trigonometric identities from the list provided are:
1. **tangent theta equals Start Fraction sine theta over cosine theta End Fraction**
2. **cosine theta equals Start Fraction 1 over secant theta End Fraction**
5. **1 plus tangent squared theta equals secant squared theta**