The First Global Age Get out something on which to take notes!
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GET OUT PAPER FOR NOTES!!!
Warm-up (3:30 m)
1. Solve for all solutions graphically: sin3x = –cos2x2. 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?
k
sin3x = –cos2x
cos x = 1
• x = 0 + 2kπ• x = 6.283 + 2kπ, k
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
k,πk2927.x927.x
)6.0(cosx
6.0xcos1
Why are there two solutions?
k,πk2356.5xπk2927.x
Let’s consider the Unit CircleWhere is x
(cosine) positive?
“All Students Take Calculus”AS
CT
all ratios are positive
sine is positive
tangent is positive
cosine is positive
cosecant is positive
cotangent is positive
secant is positive
How do we find the other solutions algebraically?
For Cosine For Sine
Calculator Solution
– Calculator Solution
Calculator Solution
π – Calculator Solution
cos x = 0.6
k,πk2927.x927.x
)6.0(cosx
6.0xcos1
πk2356.5x
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…
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
2. Explain why we would reject the solution cos x = 10
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?
• x2 + 5x + 3 = 0
a2ac4bbx
2
The plus or minus symbol means that you
actually have TWO equations!
Quadratic Formula
x2 + 5x + 3 = 0ax2 + 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:• ax2 + bx + c = 0• 2 cos2x + 6 cos x – 4 = 0• 7 tan2x + 10 = 0
tan2x + 5 tan x + 3 = 0
3 sin2x – 8 sin x = –3
Your Turn:• Solve for all solutions algebraically:
1. sin2x + 2 sin x – 2 = 0
2. tan2x – 2 tan x = 2
3. cos2x = –5 cos x + 1
Seek and Solve!
Remember me?
xtan1xcot
xsin1xcsc
xcos1xsec
xsec1xtan
xcscxcot1
1xcosxsin
22
22
22
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!
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
cos2x – sin2x + sin x = 0
sec2x – 2 tan2x = 0
sec2x + tan x = 3
Your Turn:• Use Pythagorean identities to solve for all
solutions algebraically:
1. –10 cos2x – 3 sin x + 9 = 0
2. –6 sin2x + cos x + 5 = 0
3. sec2x + 5 tan x = –2