Question

which of the following trigonometric relationships are true for right triangles?

csc(θ)=opposite/hypotenuse

cos(θ)=adjacent/hypotenuse

sin(θ)=opposite/hypotenuse

cot(θ)=cos(θ)/sin(θ)

tan(θ)=cos(θ)/sin(θ)

Explanation:

Step1: Recall trigonometric definitions

In a right - triangle, for an angle \(\theta\):

• \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\)
• \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\)
• \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\)
• \(\cot\theta=\frac{\text{adjacent}}{\text{opposite}}=\frac{\cos\theta}{\sin\theta}\) (since \(\frac{\cos\theta}{\sin\theta}=\frac{\frac{\text{adjacent}}{\text{hypotenuse}}}{\frac{\text{opposite}}{\text{hypotenuse}}}=\frac{\text{adjacent}}{\text{opposite}}\))
• \(\csc\theta=\frac{\text{hypotenuse}}{\text{opposite}}\)

Step2: Check each option

• For \(\csc(\theta)=\frac{\text{opposite}}{\text{hypotenuse}}\): Incorrect. Because \(\csc\theta=\frac{\text{hypotenuse}}{\text{opposite}}\)
• For \(\cos(\theta)=\frac{\text{adjacent}}{\text{hypotenuse}}\): Correct.
• For \(\sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}}\): Correct.
• For \(\cot(\theta)=\frac{\cos(\theta)}{\sin(\theta)}\): Correct.
• For \(\tan(\theta)=\frac{\cos(\theta)}{\sin(\theta)}\): Incorrect. Because \(\tan\theta = \frac{\sin\theta}{\cos\theta}\)

Answer:

\(\cos(\theta)=\frac{\text{adjacent}}{\text{hypotenuse}}\), \(\sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}}\), \(\cot(\theta)=\frac{\cos(\theta)}{\sin(\theta)}\)