Chapter 5: Trigonometric Functions

Lesson 7: General Solutions to Trigonometric Equations

Mrs. Parziale

Vocabulary

• General Solutions – a basic way to write an equation for all possible values (infinitely many) without listing all values.

Example 1:

• Solve for all : • Use the calculator. Graph

y = sin(x) and y = 0.4199. – Set your window to have x-

max of 4– How many solutions. How

would you write them all? – Find two key ones – How often

do they repeat? •

sin 0.4199

5 10–5–10 x

1

2

3

–1

–2

–3

y

Example 1:

• Solve for all :

• General solution are _____________________ or ____________________________

sin 0.4199

Finding the Other Angle Measure

For TRIG(x) = A (i.e. sin(x) = .4199)1. Look at A. Determine if it is positive or negative and write down

which quadrants your answers will be in.2. Take the INVERSETRIG (|A|) and write it down as the ref angle.3. Then, calculate the other angle as follows:

WRITE THIS DOWN

Quad II Quad 1Quad I

Quad III Quad IV

x ref anglex

x 2 x

Example 2:

• Solve for all in radians:

• General solutions are _____________________ or ____________________________

cos 0.158

Example 3:

• a) Find all values for (x) in radians: 23tan 4 tan 1 0x x

Example 3, cont.

• b) Find all values for in radians:

• Use calculator and write the general solution. Remember, the period of tangent is .

1tan

3

Example 3, cont.

• c) Find all values for in radians:

• Use the unit circle and write the general solution.

tan 1

Example 4:

• Find all values for (x) in radians: 2sin 2sin 1 0x x

Closure

• What is a general solution?• What is added to either a sin or cos function in

the general solution?• What is added to the tan function in the

general solution?• Solve the following:

12sin 7 0 tan 1.2985