Question

find the signs of the six trigonometric function values for the given angle.
86°

complete the table.
function\tsign
sin 86°
cos 86°
tan 86°
csc 86°
sec 86°
cot 86°

Explanation:

Step1: Determine the quadrant of \(86^\circ\)

Angles between \(0^\circ\) and \(90^\circ\) lie in the first quadrant. Since \(0^\circ< 86^\circ< 90^\circ\), \(86^\circ\) is in the first quadrant.

Step2: Recall the sign rules for trigonometric functions in the first quadrant

In the first quadrant, all trigonometric functions (sine, cosine, tangent, cosecant, secant, cotangent) are positive. This is because for any angle \(\theta\) in the first quadrant, the \(x\) - coordinate (related to cosine) and \(y\) - coordinate (related to sine) of the corresponding point on the unit circle are positive. And since \(\tan\theta=\frac{\sin\theta}{\cos\theta}\), \(\csc\theta = \frac{1}{\sin\theta}\), \(\sec\theta=\frac{1}{\cos\theta}\), \(\cot\theta=\frac{\cos\theta}{\sin\theta}\), when \(\sin\theta>0\) and \(\cos\theta > 0\), all these functions will also be positive.

For \(\sin86^\circ\): Since \(86^\circ\) is in the first quadrant, \(\sin86^\circ>0\), so the sign is positive.

For \(\cos86^\circ\): Since \(86^\circ\) is in the first quadrant, \(\cos86^\circ>0\), so the sign is positive.

For \(\tan86^\circ=\frac{\sin86^\circ}{\cos86^\circ}\): Since \(\sin86^\circ>0\) and \(\cos86^\circ>0\), \(\tan86^\circ>0\), so the sign is positive.

For \(\csc86^\circ=\frac{1}{\sin86^\circ}\): Since \(\sin86^\circ>0\), \(\csc86^\circ>0\), so the sign is positive.

For \(\sec86^\circ=\frac{1}{\cos86^\circ}\): Since \(\cos86^\circ>0\), \(\sec86^\circ>0\), so the sign is positive.

For \(\cot86^\circ=\frac{\cos86^\circ}{\sin86^\circ}\): Since \(\cos86^\circ>0\) and \(\sin86^\circ>0\), \(\cot86^\circ>0\), so the sign is positive.

Answer:

Function	Sign
\(\cos86^\circ\)	Positive
\(\tan86^\circ\)	Positive
\(\csc86^\circ\)	Positive
\(\sec86^\circ\)	Positive
\(\cot86^\circ\)	Positive