To determine which of the statements are true, we need to know the definitions of the trigonometric functions involved:
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Tangent (tan) \[ \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \]
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Cotangent (cot) \[ \cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)} \]
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Secant (sec) \[ \sec(\theta) = \frac{1}{\cos(\theta)} \]
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Cosecant (csc) \[ \csc(\theta) = \frac{1}{\sin(\theta)} \]
Now let's evaluate the statements provided:
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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.
1 answer