Question

find the signs of the six trigonometric function values for the given angle.
39°

sin 39° is positive, cos 39° is positive, and tan 39° is positive.
csc 39° is ▼, sec 39° is ▼, and cot 39° is ▼

Explanation:

Step1: Recall reciprocal relationships

The reciprocal of sine is cosecant, so \(\csc\theta=\frac{1}{\sin\theta}\). The reciprocal of cosine is secant, so \(\sec\theta = \frac{1}{\cos\theta}\). The reciprocal of tangent is cotangent, so \(\cot\theta=\frac{1}{\tan\theta}\).

Step2: Analyze sign of cosecant

Since \(\sin39^{\circ}\) is positive, and \(\csc39^{\circ}=\frac{1}{\sin39^{\circ}}\), a positive number divided by a positive number is positive. So \(\csc39^{\circ}\) is positive.

Step3: Analyze sign of secant

Since \(\cos39^{\circ}\) is positive, and \(\sec39^{\circ}=\frac{1}{\cos39^{\circ}}\), a positive number divided by a positive number is positive. So \(\sec39^{\circ}\) is positive.

Step4: Analyze sign of cotangent

Since \(\tan39^{\circ}\) is positive, and \(\cot39^{\circ}=\frac{1}{\tan39^{\circ}}\), a positive number divided by a positive number is positive. So \(\cot39^{\circ}\) is positive.

Answer:

\(\csc 39^{\circ}\) is positive, \(\sec 39^{\circ}\) is positive, and \(\cot 39^{\circ}\) is positive.