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5.1 notes: Fundamental Identities
Trig identities: are statements that are true for all values of the variable for which both sides of the equation are
defined. For example: tan θ = sin θ/cos θ or csc θ = 1/sin θ
Domain of validity: set of all true values for an identity
Basic Identities:
Reciprocal Identities:
csc θ =
sec θ =
Quotient identities:
tan θ =
cot θ =
sin θ =
cos θ =
cot θ =
Do p. 405 exploration 1
Pythagorean Identities:
cos2 θ + sin2θ = 1
1 + tan2 θ = sec2 θ
cot2 θ + 1 = csc2 θ
examples: evaluate without using a calculator. Use the Pythagorean identities
1. Find sec θ and csc θ if tan θ = 3 and cos θ >0
2. Find sin θ and tan θ if cos θ = .8 and tan θ < 0
3. Find sinθ and cosθ of tanθ = 4 and cosθ > 0.
Cofunction identities:
sin (
cos (
tan (
csc (
sec (
cot (
Odd-Even Identities:
sin (-x) = -sin x
csc (-x) = - csc x
cos (-x) = cos x
sec (-x) = sec x
tan (-x) = - tan x
cot (-x) = - cot x
examples: Use identities to find the value of the expression
4. If tan (
5. If cot (-θ) = 7.89, find tan (θ -
6. If cosθ = 0.26, find sin(θ – π/2).
Simplifying by Combining Fractions and Using Identities
7. Simplify the expression __cosx__ - _sinx_ .
1 – sinx
cos x
Examples: combine the fractions and simplify to a multiple of a power of a basic trig function
8.
1
tan θ =
9.
Simplifying by Factoring and Using Identities
Simplify the expression sin3x + sinx cos2x.
Solving a Trigonometric Equation by Factoring
Find all solutions to the trigonometric equation 2sin2x + sinx = 1.
Examples: write each expression in factored form as an algebraic expression of a single trig function
10. 1 – 2sin x + sin2 x
11. sin2 x +
Solving a Trigonometric Equation
Find all solutions to the equation in the interval [0, 2π).
12. √2 tan x cos x – tan x = 0
13. tan x sin2 = tan x
14. tan2x = 3
15. 2 sin2 x = 1
16. 2 sin2x + 3 sin x + 1 = 0
17. sin2x – 2sin x = 0
Make the suggested trigonometric substitution and then use Pythagorean identities to write the resulting
function as a multiple of a basic trigonometric function.
18.
19.
20.
, x = 3 sec
, x = 6 sin
, x = 10 sec
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Proving trigonometric Identities (5.2)
Proofs must be organized in a manner that is easy to follow by anyone. You should never include arrows
or other symbols in your proof. Each line of your proof must be equivalent to the next.
You should always try to solve a proof by working only one side. Although it is not incorrect to work
both sides of a proof in order to prove equality, there is often a much simpler and much more clear way
to solve the proof if you only use manipulations on one side. If you do choose to work both sides of a
proof, you must have each line of the proof equivalent to the other side. On your take home test you will
only be permitted to solve one proof by doing manipulations to both sides.
General Strategies I
1. The proof begins with the expression on one side of the identity.
2. The proof ends with the expression on the other side.
3. The proof in between consists of showing a sequence of expressions, each one easily seen to be
equivalent to its preceding expression.
Proving an Algebraic Identity
1. Prove the algebraic identity x2 – 1
x–1
_
x2 – 1
x+1
= 2.
General Strategies II
1. Begin with the more complicated expression and work toward the less complicated expression.
2. If no other move suggests itself, convert the entire expression to one involving sines and cosines.
3. Combine fractions by combining them over a common denominator.
Proving an Identity
2. Prove the identity: tanx + cotx = secx cscx .
3. Prove the following identity:
1
+
1
_.
secx – 1
secx + 1
General Strategies III
1. Use the algebraic identity (a + b) (a – b) = a2 – b2 to set up applications of the Pythagorean identities.
2. Always be mindful of the “target” expression, and favor manipulations that bring you closer to your
goal.
Setting up a Difference of Squares
4. Prove the identity:
cos t
1 – sin t
=
1 + sin t .
cos t
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Proving Trigonometric Identities (5.2): Classwork
Prove the identity.
1. (cosx)(tanx + sinx cotx) = sinx + cos2x
2. (cosx – sinx)2 = 1 – 2sinx cosx
3.
= -tanx sinx
4. (1 – sinx)(1 + cscx) = 1 – sinx + cscx – sinx cscx
5. cot2x – cos2x = cos2xcot2x
6. tan x + sec x =
cos x
1 – sin x
7. sec x + 1 =
tan x _
tan x
sec x – 1
5.3 notes:
Cosine of a Sum or Difference: cos (u ± v) = cos u cos v sin u sin v
Sine of a Sum or Difference: sin (u ± v) = sin u cos v ± cos u sin v
Tangent of sum or difference: sin (u ± v) = sin u cos v ± cos u sin v
cos (u ± v) = cos u cos v
OR
sin u sin v
tan (u ± v) = tan u ± tan v
1 tan u tan v
Examples:
1) use a sum or difference identity to find an exact value:
a. rewrite the given θ as a sum or difference of 2 θ’s – use 30, 45, or 60 (or an equivalent of them)
b. Rewrite your expression using the sum or difference identities
c. Solve
1. tan 15˚
2. cos 75˚
3. sin
2) write the expression as the sine, cosine, or tangent of an angle.
a. determine which sum or difference you are using, then rewrite
b. then add or subtract your angles together
4. cos 94˚ cos 18˚ + sin 94˚ sin 18˚
5. sin
6.
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Multiple Angle Identities (5.4)
Double-Angle Identities
sin 2u = 2 sin u cos u
cos 2u =
cos2u - sin2u
2 cos2u – 1
1 – 2 sin2u
tan 2u = 2 tan u
1 - tan2u
Proving a Double-Angle Identity
1. Prove the identity: cos 2u = cos2u – sin2u.
Power-Reducing Identities
sin2u = 1 – cos 2u
2
cos2u = 1 + cos 2u
2
tan2u = 1 – cos 2u_
1 + cos 2u
Proving a Power of 4
Rewrite sin4x in terms of trigonometric functions with no power greater than 1.
Proving an Identity
Prove the identity: cos4x – sin4x = cos 2x.
Exploration: Finding the Sine of Half an Angle
Recall the power-reducing formula sin2u = (1 – cos 2u)/2.
2. Use the power-reducing formula to show that sin2(π/8) = (2 - √2)/4.
3. Use the power-reducing formula to show that sin2(9π/8) = (2 - √2)/4.
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Half-Angle Identities
sin
cos
tan
Using a Double-Angle Identity
4. Solve algebraically in the interval [0, 2 π): sin 2x = cos x.
Using Half-Angle Identities
5. Solve sin2x = 2 sin2(x/2).
The Law of Sines (5.5)
Law of sines
Let ∆ABC be any triangle with angles A, B and C and sides a, b, and c then
Examples: find the missing angles and side lengths using Law of sines
1. B = 47°, a = 8, b = 21
2. A = 19°, b = 22, c = 47
3. C = 115°, b = 12, c = 7
4. A = 136°, a = 15, b = 28
The Law of Cosines (5.6)
Law of Cosines
Let ∆ABC be any triangle with angles A, B and C and sides a, b, and c then
a2 = b2 + c2 – 2bc cos A
b2 = a2 + c2 – 2ac cos A
c2 = a2 + b2 – 2ab cos A
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Solving a Triangle (SAS)
1. Solve ∆ABC given that <B = 131o, a = 13, and c = 8.
Solving a Triangle (SSS)
2. Solve ∆ABC given that a = 27, b = 19 and c = 24.
Measuring a Baseball Diamond
3. The bases on a baseball diamond are 90 feet apart, and the front edge of the pitcher’s rubber is 60.5 feet
from the back corner of home plate. Find the distance from the center of the front edge of the pitcher’s
rubber to the far corner of first base.
Area of a Triangle
∆ Area = ½ bc sin A
∆ Area = ½ ac sin B
∆ Area = ½ ab sin C
Finding the Area of a Triangle
4. Find the area of ∆ABC given that A = 47o, b = 32 ft, and c = 19 ft.
Heron’s Formula
Let a, b and c be the sides of ∆ABC, and let s denote the semiperimeter where
s = (a + b + c)/2
Then the area of ∆ABC is given by
Area =
Using Heron’s Formula
5. Find the area of a triangle with sides a = 4, b = 5 and c = 8.
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