Trigonometry Angle Questions with Answers

Trigonometry questions related to angles in standard position, coterminal angles, complementary and supplementary angles, as well as conversion from degrees to radians and vice versa, are presented. The solutions and answers are provided.

Question 1

Graph \( -435^\circ \) in standard position.

Solution

• Start from the initial side on the horizontal axis, positive direction, rotate \(435^\circ\) in the negative direction to locate the terminal side which is in quadrant four. It helps to note that \[ 435^\circ = 360^\circ + 75^\circ \]

Question 2

Graph \( \dfrac{9\pi}{4} \) in standard position.

Solution

• Start from the initial side on the horizontal axis, positive direction, rotate \( \dfrac{9\pi}{4} \) radians in the positive direction to locate the terminal side which is in quadrant one. Note that \[ \dfrac{9\pi}{4} = 2\pi + \dfrac{\pi}{4} \]

Question 3

In which quadrant is the terminal side of an angle of \( -\dfrac{3\pi}{4} \) located?

Solution

• The terminal side of \( -\dfrac{3\pi}{4} \) is located in quadrant three.

Question 4

In which quadrant is the terminal side of an angle of \(750^\circ\) located?

Solution

• \[ 750^\circ = 360^\circ + 360^\circ + 30^\circ \] Hence an angle of \(750^\circ\), in standard position, has its terminal side in quadrant one.

Question 5

Find a coterminal angle \(t\) to angle \( -\dfrac{27\pi}{12} \) such that \(0 \le t < 2\pi\).

Solution

• We first note that \[ -\dfrac{27\pi}{12} = -\dfrac{24\pi}{12} - \dfrac{3\pi}{4} = -2\pi - \dfrac{3\pi}{4} \] A coterminal angle is obtained by adding or subtracting a whole number of \(2\pi\). Hence a positive coterminal angle is obtained by adding \(4\pi\): \[ t = -\dfrac{27\pi}{12} + 4\pi = \dfrac{7\pi}{4} \]
• Note that \(t\) is positive and smaller than \(2\pi\).

Question 6

Find an angle \(t\) that is coterminal to \(560^\circ\) such that \(0 \le t < 360^\circ\).

Solution

• Since \[ 560^\circ = 360^\circ + 200^\circ \] we subtract \(360^\circ\): \[ t = 560^\circ - 360^\circ = 200^\circ \]

Question 7

Determine the complementary angle \(t\) to \( \dfrac{\pi}{12} \).

Solution

• \[ t = \dfrac{\pi}{2} - \dfrac{\pi}{12} = \dfrac{5\pi}{12} \]

Question 8

Determine the complementary angle \(t\) to \(34^\circ\).

Solution

• \[ t = 90^\circ - 34^\circ = 56^\circ \]

Question 9

Determine the supplementary angle \(t\) to \(96^\circ\).

Solution

• \[ t = 180^\circ - 96^\circ = 84^\circ \]

Question 10

Convert \(75^\circ\) to radians.

Solution

• \[ 75^\circ \times \dfrac{\pi}{180} = \dfrac{5\pi}{12} \approx 1.31 \]

Question 11

Convert \( \dfrac{7\pi}{4} \) to degrees.

Solution

• \[ \dfrac{7\pi}{4} \times \dfrac{180}{\pi} = 315^\circ \]

Question 12

Convert \(1.5\) radians to degrees.

Solution

• \[ 1.5 \times \dfrac{180}{\pi} \approx 85.94^\circ \]

Question 13

Convert \(61^\circ 05' 12''\) to degrees in decimal form.

Solution

• \[ 61 + \dfrac{5}{60} + \dfrac{12}{3600} \approx 61.07^\circ \]

Question 14

A central angle \(t\) of a circle with radius \(2\) meters subtends an arc of length \(1.5\) meters. Find angle \(t\) in degrees.

Solution

• Using the arc length formula \(s = rt\): \[ 1.5 = 2t \]
• Solving: \[ t = 0.75 \text{ radians} \]
• Converting to degrees: \[ t = 0.75 \times \dfrac{180}{\pi} \approx 42.97^\circ \]

More References and Links

Trigonometry Problems with Answers