10.24 Functions on the Unit Circle

• 0

• undefined

• 1

• -1

Answer explanation

The secant function, \sec(x), is defined as \frac{1}{\cos(x)}. At \frac{3\pi}{2}, \cos\left(\frac{3\pi}{2}\right) = 0, making \sec\left(\frac{3\pi}{2}\right) undefined.

Tags

CCSS.HSF.TF.A.1

CCSS.HSF.TF.A.2

CCSS.HSF.TF.A.3

CCSS.HSF.TF.C.8

Answer explanation

To find \( \sin\left(\frac{7\pi}{6}\right) \), note that \( \frac{7\pi}{6} \) is in the third quadrant where sine is negative. The reference angle is \( \frac{\pi}{6} \), giving \( \sin\left(\frac{7\pi}{6}\right) = -\frac{1}{2} \).

Tags

CCSS.HSF.TF.A.2

CCSS.HSF.TF.A.3

CCSS.HSF.TF.A.4

CCSS.HSF.TF.C.8

• undefined

• 0

• 1

• -1

Answer explanation

The tangent function, \( \tan(x) \), is periodic with a period of \( \pi \). Thus, \( \tan(-3\pi) = \tan(0) = 0 \). Therefore, the correct answer is 0.

Tags

CCSS.HSF.TF.A.2

CCSS.HSF.TF.A.4

CCSS.HSF.TF.C.8

Answer explanation

The angle \( \frac{3\pi}{4} \) is in the second quadrant of the unit circle. The coordinates are given by \( \left( \cos \frac{3\pi}{4}, \sin \frac{3\pi}{4} \right) = \left( -\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2} \right) \).

Tags

CCSS.HSF.TF.A.3

CCSS.HSF.TF.C.8

• I

• II

• III

• IV

Answer explanation

The angle -\frac{2\pi}{3} radians is equivalent to 240 degrees, which lies in the third quadrant where both sine and cosine values are negative. Therefore, the correct answer is III.

Tags

CCSS.HSF.TF.A.2

CCSS.HSF.TF.A.4

What is the radius of a unit circle?

• 1

• 2

Answer explanation

A unit circle is defined as a circle with a radius of 1. Therefore, the correct answer is 1, as it represents the radius of a unit circle.

Tags

CCSS.HSG.C.B.5

CCSS.HSG.GPE.A.1

Answer explanation

To find \( \csc\left(\frac{3\pi}{4}\right) \), we first find \( \sin\left(\frac{3\pi}{4}\right) = \frac{\sqrt{2}}{2} \). The cosecant is the reciprocal, so \( \csc\left(\frac{3\pi}{4}\right) = \frac{1}{\sin\left(\frac{3\pi}{4}\right)} = \sqrt{2} \).

Tags

CCSS.HSF.TF.A.3

CCSS.HSF.TF.C.8

CCSS.HSG.SRT.C.6

CCSS.HSG.SRT.C.8

• 1

Answer explanation

The cosine of π/4 (45 degrees) is ±√2/2. Since cosine is positive in the first quadrant, π/4 gives √2/2, which simplifies to \frac{\sqrt{2}}{2}. Thus, the correct answer is \frac{\sqrt{2}}{2}.

Tags

CCSS.HSF.TF.A.3

Answer explanation

To find \( \sin\left(\frac{11\pi}{6}\right) \), note that \( \frac{11\pi}{6} \) is in the fourth quadrant where sine is negative. The reference angle is \( \frac{\pi}{6} \), giving \( \sin\left(\frac{11\pi}{6}\right) = -\frac{1}{2} \).

Tags

CCSS.HSF.TF.A.2

CCSS.HSF.TF.A.3

CCSS.HSF.TF.A.4

CCSS.HSF.TF.C.8

What is the reciprocal of the sine function?

• cosecant

• cosine

• secant

• cotangent

Answer explanation

The reciprocal of the sine function is called cosecant. It is defined as cosecant(x) = 1/sin(x). The other options, cosine, secant, and cotangent, represent different trigonometric functions.