Warm up Solve for the missing side length
Essential Question: How to right triangles relate to the unit circle? How can I use special triangles to find coordinates on the unit circle?
Standard: MM4A2. Students will use the circle to define the trigonometric functions.
c. Find values of trigonometric functions using points on the terminal sides of angles in the standard position.
e. Find values of trigonometric functions using the unit circle.
Math IV Lesson 22
Review of the Pythagorean theorem
Using the Pythagorean theorem
9
The trigonometric functions are
sine, cosine, tangent, cotangent, secant, and cosecant.
opp
adj
hyp
θ
sin = cos = tan =
csc = sec = cot = opp
hypadjhyp
hypadj
adjopp
oppadj
hypopp
Note: sine and cosecant are reciprocals, cosine and secant are reciprocals and tangent and cotangent are reciprocals
The Trigonometric Functionson the unit circle (we learned these last Thursday)
Let t be a real number and let (x,y) be the point on the unit circle corresponding to t
Sin(t) = y csc(t) = 1/y
Cos(t) = x sec(t) = 1/x
Tan(t) = y/x cot(t) = x/y
A Unit Circle has Radians, degrees and coordinates
A circle defined by x2 + y2 = 1
(x, y)
12
The Unit Circle
• Imagine the real number line wrapped around the circle.
• Each real number t corresponds to a point (x, y).
• Since the radius is 1, the number t would correspond with the central angle (s = rθ).
(1, 0)
(0, -1)
(-1, 0)
(0, 1)
t
θ= t
13
Geometry of the 45-45-90 triangle
Consider an isosceles right triangle with a hypotenuse the length of 1.
22
22
45
451 What would be the length of the sides?
2245cos
2245sin oo
14
21
23
Geometry of the 30-60-90 triangle
1
30
60○
2160cos
2360sin
2330cos
2130sin
oo
oo
Consider a 30-60-90 triangle with a hypotenuse the length of 1.
What would be the length of the sides?
Signs on the unit circle
Use special right triangles to fill in the coordinates on the unit circle
Evaluating trigonometric functions using special triangles
Solve each triangle. Redraw the triangles here and write in the lengths of the sides. a. 120 degrees b. 135 degrees c. 150 degrees
Evaluate each function without using a calculator. (Draw special right triangles in position on the Unit Circle and apply the Unit Circle Definition
of the trigonometric functions.)
• 1. Sin(240 degrees) • 2. cos(315 degrees)
Wednesday
• Quiz on the coordinates of the unit circle!!!!