Trigonometry: Solving Trigonometric Equations
Trigonometry: Solving Trigonometric Equations 5/31/2010
Pure Math 20
Previously:
We learned the equations for the reference angles of each quadrant.
[pic]
We also learned the CAST rule for finding where trig ratios are positive.
[pic]
These two strategies enabled us to solve equations like the following:
a. cos x = 0.8245 b. sin x = -0.7986
2nd cos(.8245) = 34( 2nd sin(0.7986) = 53( reference angle
Since cos is also positive in Q4, use Sin is negative in Q3 and Q4
360 – x. So 360 – 34 = 326( x = 180 + 53 = 233( and
x = 34( and 326( x = 360 – 53 = 307(
c. tan m = -3.0777
2nd tan(3.0777) = 72(
The reference angle is 72(
Tan is negative in Q2 and Q4
m = 180 – 72 = 108(
m = 360 – 72 = 288(
Now solve the following questions for (.
a. sin ( = 0.6128 b. tan ( = 1.5738
c. cos ( = -0.8175 d. sin ( = ¾
e. tan ( = 1/(3 f. cos ( = -1/(2
g. cos ( = (3/2 h. sin ( = - 1
i. tan ( = -3(5
Solving More Complex Equations
Example: Solve the following;
a. 3cos x – 1 = 0
The goal is to isolate the trig ratio.
3cosx = 1
cos x = 1/3
Now, this looks like something we can solve.
2nd cos(1/3) = 70.5(
and 360 – 70.5 = 289.5(
b. 3sin x – 4 = 11sin x
3sin x – 11sin x = 4
-8sin x = 4
sin x = -4/8
sin x = -½
2nd sin( ½ ) = 30 (reference angle)
180 + 30 = 210(
360 – 30 = 330(
c. 3tan x – 1 = 4
3tan x = 5
tan x = 5/3
2nd tan(5/3) = 59(
x = 180 + 59 = 239(
Now do these from the handout.
Handout: Exercises 10–8 #3odds, 4odds, 5a – d, 8a – d, 9a – d, 12a – b
Extension: Problem Solving:
The point (3, - 2) lies on the terminal arm of an angle, m, in standard position. Find the value of m to the nearest degree.
1. Draw diagram to see the information.
[pic]
2. The angle is in quadrant 4. To solve the appropriate trig ratio, we must use the value of the point (3, -2) as the lengths of the opposite and adjacent sides of a right triangle.
[pic]
3. Now solve for the angle using tan m = o/a, which is tan m = -2/3.
2nd tan(2/3) = 34 reference angle
360 – 34 = 326(
Note: there is only one answer because the diagram defines that it only occurs in quadrant 4.
Now try Handout Exercises 10–8 #6
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