An angle \theta in standard position is coterminal with an angle of -\frac{17\pi}{4} radia… — ACT

An angle $\theta$ in standard position is coterminal with an angle of $-\frac{17\pi}{4}$ radians. If $0 \le \theta < 2\pi$ , what is the value of $\theta$ expressed as a decimal multiple of $\pi$ ? (For example, if the angle were $\frac{3\pi}{2} = 1.5\pi$ , the answer would be 1.5.)

Answer: 1.75

Answer

1.75

To find a coterminal angle in the interval

[0, 2\pi)

for

-\frac{17\pi}{4}

, we add multiples of

2\pi

. Since

-\frac{17\pi}{4} = -4.25\pi

, we add

6\pi

(three full rotations) to get

-4.25\pi + 6\pi = 1.75\pi

. The multiple of

\pi

is therefore 1.75.

Step-by-Step Solution

Convert the coefficient of the given angle from a fraction to a decimal.

-\frac{17}{4} = -4.25

Converting the fraction to a decimal makes it easier to work with the addition of full rotations.

Add multiples of

2\pi

(which corresponds to adding 2 to the coefficient of

\pi

) to find a coterminal angle in the interval

[0, 2\pi)

-4.25 + 2 = -2.25

;

-2.25 + 2 = -0.25

;

-0.25 + 2 = 1.75

Adding

2\pi

representing full counterclockwise rotations on the unit circle results in a coterminal angle. We repeat this process until the coefficient lies in the interval

[0, 2)

Identify the coefficient of

\pi

for the coterminal angle.

1.75

The question asks for the angle as a decimal multiple of

\pi

, which is the coefficient of

\pi

in the expression

1.75\pi

Key Concept

Coterminal angles are angles in standard position that share the same terminal side. They can be found by adding or subtracting multiples of

2\pi

radians.

Estimated Time:1m 30s

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