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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