Question

find the signs of the six trigonometric function values for the given angle.
299°

sin 299° is negative, cos 299° is positive, and tan 299° is negative.
csc 299° is ▼ sec 299° is ▼ and cot 299° is ▼

Explanation:

Step1: Recall reciprocal identities

The reciprocal identities are: \(\csc\theta=\frac{1}{\sin\theta}\), \(\sec\theta = \frac{1}{\cos\theta}\), \(\cot\theta=\frac{1}{\tan\theta}\) (or \(\cot\theta=\frac{\cos\theta}{\sin\theta}\)).

Step2: Analyze \(\csc299^{\circ}\)

We know that \(\sin299^{\circ}\) is negative. Since \(\csc299^{\circ}=\frac{1}{\sin299^{\circ}}\), the reciprocal of a negative number is negative. So \(\csc299^{\circ}\) is negative.

Step3: Analyze \(\sec299^{\circ}\)

We know that \(\cos299^{\circ}\) is positive. Since \(\sec299^{\circ}=\frac{1}{\cos299^{\circ}}\), the reciprocal of a positive number is positive. So \(\sec299^{\circ}\) is positive.

Step4: Analyze \(\cot299^{\circ}\)

We know that \(\tan299^{\circ}\) is negative. Since \(\cot299^{\circ}=\frac{1}{\tan299^{\circ}}\), the reciprocal of a negative number is negative. Also, using \(\cot\theta=\frac{\cos\theta}{\sin\theta}\), \(\cos299^{\circ}\) is positive and \(\sin299^{\circ}\) is negative, so positive divided by negative is negative. So \(\cot299^{\circ}\) is negative.

Answer:

\(\csc 299^{\circ}\) is negative, \(\sec 299^{\circ}\) is positive, and \(\cot 299^{\circ}\) is negative.