10-3 Double-Angle and Half-Angle

Double-angle Formulas:

Sin2α = 2sinαcosα cos2α = cos^2α-sin^2α cos2α = 2cos^2α-1 cos2α = 1-2sin^2α

Half-angle Formulas:

Tan2α = +/- 2tanα/1-tan^2α sinα/2 = +/- sqrt1-cosα/2 cosα/2 = +/- sqrt1+cosα/2

Tanα/2 = +/- sqrt1-cosα/1+cosα tanα/2 = sinα/1+cosα , 1-cosα/sinα

Ex. If sinα = 4/5 and 0<α

Sin2α =2sinαcosα

Already have sin so replace it – sin2α = 2(4/5)cosα

To find cosα just draw the unit circle and find out which quadrant it’s in with 0<α

Draw a triangle with that and it’s a 3, 4, 5 triangle. Cosα= 3/5

Sin2α = 2(4/5)(3/5) = 24/25

Cos2α = 1-2sin^2α = 1-2(4/5)^2 = -7/25

Go back to your triangle and find out that tanα = 4/3

Tan2α = 2tanα/1-tan^2α = 2(4/3)/1-(4/3)^2 = 8/3/-7/9 = -72/21

Double- and Half-Angle Formulas

Here they are:

sin2(theta)=2sin(theta)cos(theta)

cos2(theta)=cos^2(theta)-sin^2(theta)

cos2(theta)=1-2sin^2(theta)

cos2(theta)=2cos^2(theta)-1

tan2(theta)=2tan(theta)/1-tan^2(theta)

sin theta/2=+/-square root of 1-cos(theta)/2

cos theta/2=+/-square root of 1+cos(theta)/2

tan theta/2=+/-square root of 1-cos(theta)/1+cos(theta)

tan theta/2=sin(theta)/1+cos(theta)

tan theta/2=1-cos(theta)/sin(theta)

Example:

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Simplify.

cos^2(pi/12) - sin^2(pi/12)

=cos2(pi/12)

=cos2(15)

=cos30 degrees

=sqrt of 3/2

Sum and Difference

Sum and Difference Formulas for Sine and Cosine 

                Cos(α+β)=cosαcosβ-sinαsinβ

                Cos(α-β)=cosαcosβ+sinαsinβ

                Sin(α+β)=sinαcosβ+cosαsinβ

                Sin(α-β)=sinαcosβ-cosαsinβ

To set up sum and differences, we can use special values of 30˚, 60˚, and 45˚

1)       Find the exact value of cos 15˚

Cos15˚=cos(45˚-30˚)                               Replace 15 with 45-30

=cos45cos30+sin45sin30                       plug values into the formula

=(√2/2)(√3/2)+(√2/2)(1/2)                     replace special values√

=(√6/4)+(√2/4)                                         Simplify

=(√6+√2)/4

2)      Find exact value of cos 75˚

Cos75=cos(45+30)                                  Replace 15  with 45+30

=cos45cos30-sin45sin30                       plug values into the formula

=(√2/2)(√3/2)-(√2/2)(1/2)                     replace special values

=(√6/4)-(√2/4)                                         Simplify

=(√6-√2)/4

3)      Find exact value of sin 75˚

Sin75=sin(45+30)                                    Replace 75 with 45+30

=sin45cos30+cos45sin30                       plug values into the formula

=(√2/2)(√3/2)+(√2/2)(1/2)                     replace special values

=(√6/4)+(√2/4)                                         Simplify

=(√6+√2)/4

Jenna Roussel

10-3!

Double-Angle Formulas:

~sin2(alpha)=2sin(alpha)cos(alpha)

~cos2(alpha)=cos^2(alpha)-sin^2(alpha)

~cos2(alpha)=1-sin^2(alpha)

~cos2(alpha)=2cos^2(alpha)-1

~tan2(alpha)=2tan(alpha)/1-tan^2(alpha)

Half-Angle Formulas:

~sin(alpha)/2=+/-sqrt. of 1-cos(theta)/2

~cos(alpha)/2=+/-sqrt. of 1+cos(alpha)/2

~tan(alpha)/2=+/-sqrt. of 1-cos(alpha)/1+cos(alpha)

~tan(alpha)/2=sin(alpha)/ 1+cos(alpha)

~tan(alpha)/2=1-cos(alpha)/ sin(alpha)

You will use the formulas for all problems in section 10-3.

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Ex.1: Simplify:

~2sin(40)cos(40)

sin2(40)= sin80

Ex.2: Simplify:

~1-cos(10)/ sin(10)

tan(10)/2= tan5

Ex.3: SImplify:

~2cos^2(15)-1

cos2(15)= cos30

=sqrt. of 3/2

Formulas for Tanget

Just like the addition and subteraction formulas, half angle formulas, double angel formulas there are special formulas for tanget. They are very similar in solving compared to the other formulas. You can identify a tanget formula by the word tan in the problem. Many times you are given a theta (@) and beta (B) with it because the formula uses both when solving. If you are given theta and beta you simply plug it in and solve the formula. There are two different formulas you can use, one for addition and one for subtraction.

tan (@+B) = tan@+tanB/1-tan@tanB

tan(@-B) = tan@-tanB/1=tan@tanB

Ex1 Suppose tan@=1/2 and tanB=1/3 find tan(@+B)
a.use addition formula tan@+tanB/1-tan@tanB plug in 1/2+1/3/1-(1/2)(1/3)
b.simplify 2/6+3/6/1-1/6 =1

10-2!!!!!:)

This week we learned 10-2! It was the easiest section yet because they only had two formulas! MWAHAHAH! :D In these formula's the top sign is the same as what you are doing and the bottom is the opposite.

Formulas:
tan(a+b) = tan(a)+tan(b)/1-tan(a)tan(b)

tan(a-b) = tan(a)-tan(b)/1+tan(a)tan(b)

Ex. 1.) Find tan (a + b) If a is 3/4 and b is 1/3.
So it would be 3/4 + 1/3/1-3/4*1/3
you have to make the top two numbers have the same deno, so multiply the four by three and the three by four. you would get 9/12 and 4/12. Then you have to multiply straight across for the bottom. so its 3/12. 1 is the same as 12/12, so it's 12/12-3/12=9/12.
Then its 13/12/9/12. so then you sandwich it. 12*9 and 13*12. so it's 156/108. then you have to simplify it. so it's 13/9! :)

-----KAYLA :)

Double and half angle formulas

The double and half angle formulas are as follows:

sin2(theta)=2sin(theta)cos(theta)

cos2(theta)=cos^2(theta)-sin^2(theta)

cos2(theta)=1-2sin^2(theta)

cos2(theta)=2cos^2(theta)-1

tan2(theta)=2tan(theta)/1-tan^2(theta)

sintheta/2=+/-square root of 1-cos(theta)/2

costheta/2=+/-square root of 1+cos(theta)/2

tantheta/2=+/-square root of 1-cos(theta)/1+cos(theta)

tantheta/2=sin(theta)/1+cos(theta)

tantheta/2=1-cos(theta)/sin(theta)

Example 1:

Simplify the given expression.

cos^2(pi/12)-sin^2(pi/12)

=cos2(pi/12)

=cos2(15)

=cos30degrees

=square root of 3/2

Example 2:

angleA is an acute angle.

If sinA=5/13, find sin2A and cos2A

x^2+5^2=13^2

x^2+25=169

x^2=144

x=12

cosA=12/13 tanA=5/12

sin2A=2(5/13)(12/13)

sin2A=120/169

cos2A=1-2(5/13)^2

=1-2(25/169)

=1-50/169

cos2A=119/169

Example3:

angleA is an acute angle.

If tanA=4/2, find cos2A and tan2A

2^2+1^2=r^2

4+1=r^2

5=r^2

r=square root of 5

sinA=square root of 5/5 cosA=2square root of 5/5

cos2A=1-2(square root of 5/5)^2

=1-2(5/25)=1-10/25

=15/25

cos2A=3/5

tan2A=2(1/2)/1-(1/2)^2=2/2=1/1-2/4

=1/1-1/2=1/1/2

tan2A=2