Trigonometry in the Coordinate Plane part 2
Instructor Katherine Cliff
Warm up:
Simplify if possible
\[\frac{x+6x^2}{4+3x}\]
What is the exact value of \(\sin\left(\frac{\pi}{6}\right)\)?
Examples
Find the reference angle for each angle in standard position.
\(135^\circ\)
A. \(315^\circ\)
B. \(495^\circ\)
C. \(45^\circ\)
D. \(225^\circ\)
\(1020^\circ\)
A. \(30^\circ\)
B. \(300^\circ\)
C. \(120^\circ\)
D. \(60^\circ\)
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. \(\frac{\pi}{6}\)
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:
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 value of \(\sin(\theta)\).
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:
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. \(-\frac{5}{13}\)
Computing exact sines and cosines for special angles in the coordinate plane
Find coterminal angle
Find quadrant
Find reference
Use 2 and 3 to find sine or cosine value
Example: Compute \(\cos(585^\circ)\) exactly.
Example: Compute \(\sin\left(-\frac{4\pi}{3}\right)\) exactly.
