Transcript M.A. L 5.3 Trigonometric Functions on the Unit Circle

Section 5.3
Trigonometric Functions on the
Unit Circle
Standard position
Unit Circle
Given a unit circle, the radius or
hypotenuse = 1
Sine and Cosine
MEMORIZE THIS:
The right angle is always on the x axis.
The acute angle is always at the origin.
MEMORIZE THIS TOO:
The ordered pair of the point where the terminal
side of the angle intersects the circle is (x, y)
where cosine Ѳ = x and sine Ѳ = y.
45° – 45° – 90° Triangle
s 2
s
s
30° – 60° – 90° Triangle
MEMORIZE THE TRIG RATIOS FOR THE
SPECIAL RIGHT TRIANGLES IN THE
FIRST QUADRANT
These ratios are repeated in each quadrant
around the circle, with sign changes.
Signs of Trigonometric Functions in
each Quadrant
+
+
-
+
+
-
+
+
-
+
+
-
Signs of the trig functions in
the 4 quadrants
• Quadrant I = All (are positive)
• Quadrant II = Students (Sine & Cosecant are +)
• Quadrant I = Take (Tangent & Cotangent are +)
• Quadrant I = Calculus (Cosine & Secant are +)
S
A
T
C
Use the unit circle to find the value for
the six trigonometric functions for a
135° angle.
Tan 210°
Sin 315°
Cos 225°
Consider an angle with a point on its
terminating side of (5, -12). That would be in
the 4th quadrant.
Find the values of the six trigonometric
functions for an angle Ѳ in standard position if
a point with the coordinates (-15,20) lies on its
terminal side.
If you know the value of one of the trig. functions and the quadrant in
which the terminal side of Ѳ lies, you can find the values of the
remaining 5 functions.
Suppose Ѳ is an angle in standard position
whose terminal side lies in Quadrant IV. If
29
sec  
5
Find the values of the five remaining functions
of Ѳ.
Now try these on page 296 #1-13:
1. Why is csc 1800 undefined?
2. Show that the value of sin Ѳ increases from 00 to
900 and decreases from 900 to 1800 .
cos 
3. Confirm that cot  
sin 
4. Complete the chart for the signs of the trig
functions in each quadrant.
Function
Sin α & Cos α
Cos α & Sec α
Tan α & Cos α
Quadrant. I Quadrant. II
Quadrant. III
Quadrant. IV
Use your unit circle to find the exact measure for
each of the following.
1.
2.
3.
4.
5.
6.
7.
8.
Tan 1800
Sec -900
Tan 450
Cot 2700
Tan 1350
Csc 2700
Tan 3600
Sec 1800
Find the values of the six trig functions for an angle θ in
standard position if a point with the given coordinates lies on
its terminal side.
1. (3,5)
2. (-6,6)
Use the unit circle to find the
sin (-90°)
Use the unit circle to find the
cot (270°)
Use the unit circle to find the value for
the six trigonometric functions for a
210° angle.