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Get out paper for notes!!!. Warm-up (3:30 m). Solve for all solutions graphically: sin 3 x = –cos 2 x Molly found that the solutions to cos x = 1 are x = 0 + 2kπ AND x = 6.283 + 2kπ, . Is Molly’s solution correct? Why or why not?. sin 3 x = –cos 2 x. cos x = 1. x = 0 + 2kπ
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Warm-up (3:30 m) • Solve for all solutions graphically: sin3x = –cos2x • Molly found that the solutions to cos x = 1 are x = 0 + 2kπ AND x = 6.283 + 2kπ, . Is Molly’s solution correct? Why or why not?
cos x = 1 • x = 0 + 2kπ • x = 6.283 + 2kπ,
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.
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 all ratios are positive cosecant is positive T C cosine is positive tangent is positive cotangent is positive secant is positive
Your Turn: • Solve for all solutions algebraically: cos x = – 0.3
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… 3x2 – x = 2
So what if we have… 3 sin2x – sin x = 2
What about… tan x cos2x – tan x = 0
Your Turn: • Solve for all solutions algebraically: 5. 4 sin2x = 5 sin x – 1 6. cos x sin2x = cos x 7. sin x tan x = sin x 8. 5 cos2x + 6 cos x = 8
Warm-up (4 m) 1. Solve for all solutions algebraically: 3 sin2x + 2 sin x = 5 • Explain why we would reject the solution cos x = 10
What happens if you can’t factor the equation? Quadratic Formula • x2 + 5x + 3 = 0 The plus or minus symbol means that you actually have TWO equations!
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: • ax2 + bx + c = 0 • 2 cos2x + 6 cos x – 4 = 0 • 7 tan2x + 10 = 0
Your Turn: • Solve for all solutions algebraically: • sin2x + 2 sin x – 2 = 0 • tan2x – 2 tan x = 2 • cos2x = –5 cos x + 1
Warm-up (4 m) • Solve for all solutions algebraically: • tan x cos x + 3 tan x = 0 • 2 cos2x + 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.
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, sine, and tangent!
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
Your Turn: • Use Pythagorean identities to solve for all solutions algebraically: • –10 cos2x – 3 sin x + 9 = 0 • –6 sin2x + cos x + 5 = 0 • sec2x + 5 tan x = –2