Trigonometry in the Coordinate Plane

Instructor Katherine Cliff

Warm up

plickers

  1. Find two angles that are coterminal with \(130^\circ\).

  2. Which of the following is the exact value of \(\sin\left(\frac{\pi}{6}\right)\)?

A. \(\frac{1}{2}\)

B. \(\frac{\sqrt{2}}{2}\)

C. \(\frac{\sqrt{3}}{2}\)

D. 0

  1. If we graph the point \((-3,5)\), what quadrant does it fall in?

A. QI

B. QII

C. QIII

D. QIV

Angles in the coordinate plane








Examples

For the given angle in standard position, what quadrant will its terminal side fall in?

  1. \(135^\circ\)

A. QI

B. QII

C. QIII

D. QIV

Examples

For the given angle in standard position, what quadrant will its terminal side fall in?

  1. \(660^\circ\)

A. QI

B. QII

C. QIII

D. QIV

Examples

For the given angle in standard position, what quadrant will its terminal side fall in?

  1. \(\frac{7\pi}{6}\)

A. QI

B. QII

C. QIII

D. QIV

Examples

For the given angle in standard position, what quadrant will its terminal side fall in?

  1. \(-\frac{5\pi}{3}\)

A. QI

B. QII

C. QIII

D. QIV

Reference Angles

Definition:







Reference Angles by Quadrant

Quadrant I





Quadrant II





Reference Angles by Quadrant

Quadrant III





Quadrant IV





Examples

Find the reference angle for each angle in standard position.

plickers

\(135^\circ\)

A. \(315^\circ\)

B. \(495^\circ\)

C. \(45^\circ\)

D. \(225^\circ\)

\(1020^\circ\)

A. \(315^\circ\)

B. \(300^\circ\)

C. \(45^\circ\)

D. \(225^\circ\)

Examples

Find the reference angle for each angle in standard position.

\[\frac{7\pi}{6}\]

A. \(\frac{19\pi}{6}\)

B. \(\frac{\pi}{6}\)

C. \(\frac{13\pi}{6}\)

D. \(\frac{5\pi}{6}\)

\[-\frac{8\pi}{3}\]

A. \(-\frac{2\pi}{3}\)

B. \(\frac{4\pi}{3}\)

C. \(\frac{\pi}{3}\)

D. \(\frac{\pi}{3}\)

Reference angle theorem













Where is each trig function positive?

Example 1:

The terminal side of an angle \(\theta\) in standard position passes through the point (-3, -5). Calculate the values of the six trigonometric functions for \(\theta\).

The terminal side of an angle \(\theta\) in standard position passes through the point (-3, -5). Calculate the values of the six trigonometric functions for \(\theta\).

Example 2:

plickers

The terminal side of an angle \(\theta\) in standard position passes through the point (-2, 4). Calculate the values of the six trigonometric functions for \(\theta\).

Example 2:

The terminal side of an angle \(\theta\) in standard position passes through the point (-2, 4). Calculate the values of the six trigonometric functions for \(\theta\).

A. \(\frac{2}{\sqrt{5}}\)

B. \(-\frac12\)

C. \(-2\)

D. \(-\frac{1}{\sqrt{5}}\)

Example 3:

If \(\sin(\theta) = -\frac17\) and the terminal side of angle \(\theta\) falls in QIV, find the cosine of angle \(\theta\).

Example 4:

plickers

If \(\cos(\theta) = -\frac{12}{13}\) and the terminal side of \(\theta\) falls in QIII, find the tangent of angle \(\theta\).

Example 4:

If \(\cos(\theta) = -\frac{12}{13}\) and the terminal side of \(\theta\) falls in QIII, find the tangent of angle \(\theta\).

A. \(-\frac{5}{12}\)

B. \(\frac{5}{12}\)

C. \(\frac{5}{13}\)

D. \(-\frac{5}{13}\)