GET OUT PAPER FOR NOTES Warmup 3 30

GET OUT PAPER FOR NOTES!!!

Warm-up (3: 30 m) 1. Solve for all solutions graphically: sin 3 x = –cos 2 x 2. Molly found that the solutions to cos x = 1 are x = 0 + 2 kπ AND x = 6. 283 + 2 kπ, . Is Molly’s solution correct? Why or why not?

sin 3 x = –cos 2 x

Solving Trigonometric Equations Algebraically

Inverse Trigonometric Functions • Remember, your calculator must be in RADIAN mode. • cos x = 0. 6 – We can use inverse trig functions to solve for x.

Check the solution graphically

Why are there two solutions? Let’s consider the Unit Circle Where is x (cosine) positive?

“All Students Take Calculus” S A sine is positive cosecant is positive all ratios are positive T tangent is positive cotangent is positive C cosine is positive secant is positive

How do we find the other solutions algebraically? For Cosine For Sine Calculator Solution – Calculator Solution π – Calculator Solution

cos x = 0. 6

Your Turn: • Solve for all solutions algebraically: cos x = – 0. 3

sin x = – 0. 75

Your Turn: • Solve for all solutions algebraically: sin x = 0. 5

What about tangent? • The solution that you get in the calculator is the only one! tan x = – 5

Your Turn: • Solve for all solutions algebraically: 1. cos x = – 0. 2 2. sin x = – ⅓ 3. tan x = 3 4. sin x = 4

What’s going on with #4? • sin x = 4

How would you solve for x if… 3 x 2 – x = 2

So what if we have… 3 sin 2 x – sin x = 2

What about… tan x cos 2 x – tan x = 0

Your Turn: • Solve for all solutions algebraically: 5. 4 sin 2 x = 5 sin x – 1 6. cos x sin 2 x = cos x 7. sin x tan x = sin x 8. 5 cos 2 x + 6 cos x = 8

Warm-up (4 m) 1. Solve for all solutions algebraically: 3 sin 2 x + 2 sin x = 5 2. Explain why we would reject the solution cos x = 10

3 sin 2 x + 2 sin x = 5

Explain why we would reject the solution cos x = 10

What happens if you can’t factor the equation? • x 2 + 5 x + 3 = 0 Quadratic Formula The plus or minus symbol means that you actually have TWO equations!

x 2 + 5 x + 3 = 0 ax 2 + bx + c = 0

Using the Quadratic Equation to Solve Trigonometric Equations • You can’t mix trigonometric functions. (Only one trigonometric function at a time!) • Must still follow the same basic format: • ax 2 + bx + c = 0 • 2 cos 2 x + 6 cos x – 4 = 0 • 7 tan 2 x + 10 = 0

tan 2 x + 5 tan x + 3 = 0

3 sin 2 x – 8 sin x = – 3

Your Turn: • Solve for all solutions algebraically: 1. sin 2 x + 2 sin x – 2 = 0 2. tan 2 x – 2 tan x = 2 3. cos 2 x = – 5 cos x + 1

Warm-up (4 m) • Solve for all solutions algebraically: 1. tan x cos x + 3 tan x = 0 2. 2 cos 2 x + 7 cos x – 1 = 0

tan x cos x + 3 tan x = 0

2 cos 2 x + 7 cos x – 1 = 0

Seek and Solve! You have 30 m to complete the seek and solve. Show all your work on a sheet of paper because I’m collecting it for a classwork grade.

Remember me?

Using Reciprocal Identities to Solve Trigonometric Equations • Our calculators don’t have reciprocal function (sec x, csc x, cot x) keys. • We can use the reciprocal identities to rewrite secant, cosecant, and cotangent in terms of cosine, and tangent!

csc x = 2 csc x = ½

cot x cos x = cos x

Your Turn: • Use the reciprocal identities to solve for solutions algebraically: 1. cot x = – 10 2. tan x sec x + 3 tan x = 0 3. cos x csc x = 2 cos x

Using Pythagorean Identities to Solve Trigonometric Equations • You can use a Pythagorean identity to solve a trigonometric equation when: – One of the trig functions is squared – You can’t factor out a GCF – Using a Pythagorean identity helps you rewrite the squared trig function in terms of the other trig function in the equation

2 cos x – 2 sin x + sin x = 0

2 sec x – 2 2 tan x =0

2 sec x + tan x = 3

Your Turn: • Use Pythagorean identities to solve for all solutions algebraically: 1. – 10 cos 2 x – 3 sin x + 9 = 0 2. – 6 sin 2 x + cos x + 5 = 0 3. sec 2 x + 5 tan x = – 2