Question

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

sin 39° is ▼ cos 39° is ▼ and tan 39° is ▼

Explanation:

Step1: Determine the quadrant of \(39^\circ\)

Angles between \(0^\circ\) and \(90^\circ\) lie in the first quadrant. Since \(0^\circ< 39^\circ< 90^\circ\), \(39^\circ\) is in the first quadrant.

Step2: Recall the sign rules in the first quadrant

In the first quadrant, all trigonometric functions (sine, cosine, tangent, cosecant, secant, cotangent) are positive. For sine, cosine, and tangent specifically:

• The sine function (\(\sin\theta\)) represents the y - coordinate on the unit circle. In the first quadrant, \(y>0\), so \(\sin39^\circ>0\).
• The cosine function (\(\cos\theta\)) represents the x - coordinate on the unit circle. In the first quadrant, \(x > 0\), so \(\cos39^\circ>0\).
• The tangent function is defined as \(\tan\theta=\frac{\sin\theta}{\cos\theta}\). Since both \(\sin\theta>0\) and \(\cos\theta>0\) in the first quadrant, their quotient \(\tan\theta=\frac{\sin\theta}{\cos\theta}>0\).

Answer:

\(\sin39^{\circ}\) is positive, \(\cos39^{\circ}\) is positive, and \(\tan39^{\circ}\) is positive.