Download Chapter 3 Section 3.1 and 3.2 Angles in Standard Position

3.1 & 3.2 Angles and Trigonometric Ratios
In a coordinate system, an angle 𝜃 is said to be in standard position if its vertex is at the________ and
its initial side coincides with the positive ________.
Angles in standard position that have the same ______________ side are called coterminal angles.
Example 1:
If 𝜃 = 140° in standard position, find two positive and two negative angles that are co-terminal with 𝜃.
Example 2:
Find the smallest positive coterminal angle for
a) 2595°
b) −1870°
For angle 𝜃 in standard position, the reference angle is the positive acute angle 𝜃𝑟 that is formed with
the terminal side of 𝜃 and the x-axis.
Example 3:
Find the reference angle for
a) 220°
b) 314°
c) −190°
Example 4:
Determine the smallest positive angle in quadrants I, II, III, and IV that has a reference angle of 45°
Trigonometric Ratios
𝑦
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
𝑠𝑖𝑛𝜃 = 𝑟 (ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒)
𝑐𝑜𝑠𝜃 =
𝑥
𝑟
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡
(ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒)
𝑡𝑎𝑛𝜃 =
𝑦
,𝑥
𝑥
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
≠ 0 (𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 )
Example 5: Identify the quadrant(s) for the angles satisfying the given conditions.
𝑐𝑜𝑠𝛽 < 0, 𝑡𝑎𝑛𝛽 > 0
Example 6: A point 𝑃 on the terminal side of 𝜃 is shown. Evaluate the three trigonometric functions of 𝜃.
Example 7: If 𝜃 is in standard position and the given point is on the terminal side of 𝜃, find the values of
𝑠𝑖𝑛𝜃, 𝑐𝑜𝑠𝜃 𝑎𝑛𝑑 𝑡𝑎𝑛𝜃.
a) (−2√3, 2)
b) (−3, √3)
Example 8: An equation with a restriction on 𝑥 is given. This is an equation of the terminal side of an
angle in standard position. Draw the smallest positive angle 𝜃, and find the values of
𝑠𝑖𝑛𝜃, 𝑐𝑜𝑠𝜃, 𝑎𝑛𝑑 𝑡𝑎𝑛𝜃.
3
𝑦 = − 𝑥, 𝑥 ≤ 0
2
Example 9: The value of one of the trigonometric functions is given, along with some additional
information. Use this information to find the other two trigonometric functions of 𝜃.
a) 𝑡𝑎𝑛𝜃 =
7
,𝜃
24
b) 𝑠𝑖𝑛𝜃 =
√15
4
In quadrant III
Example 10: The value of one of the trigonometric functions is given along with some additional
information. Use the trigonometric ratios to find the other two trigonometric functions of 𝜃. Round each
answer to three decimal places.
a) 𝑐𝑜𝑠𝜃 = −0.378, 𝑠𝑖𝑛𝜃 > 0
b) 𝑠𝑖𝑛𝜃 = −0.753, 𝑡𝑎𝑛𝜃 > 0