7.3­7.4_Sum_Diff_Double_Half
March 21, 2012
Finding exact cosine function values:
Cos(A + B) = cos A cos B ­ sin A sin B
Cos(A ­ B) = cos A cos B + sin A sin B
Find: cos (­750) = cos(­450 + ­ 300)
= cos (­45)0cos(­300) ­ sin(­450)sin(­300)
= √2(√3) - (-√2)(-1)
2 2 2 2
= √6 - √2
4
1
7.3­7.4_Sum_Diff_Double_Half
March 21, 2012
Finding exact cosine function values:
Cos(A + B) = cos A cos B ­ sin A sin B
Cos(A ­ B) = cos A cos B + sin A sin B
Find: cos 1730 cos830 + sin 1730 sin 830
= cos(1730 ­ 830)
= cos 900
= 0
2
7.3­7.4_Sum_Diff_Double_Half
March 21, 2012
Finding exact cosine function values:
Cos(A + B) = cos A cos B ­ sin A sin B
Cos(A ­ B) = cos A cos B + sin A sin B
Hint:
π = 2π
6
12
π = 3π
4
12
π = 4π
3
12
FIND:
cos 17π can be rewritten as -7π
12
12
-7π = -3π + -4π = -π + -π
12
12
12
4
3
= √2(1) - (-√2)(-√3)
22
2
2
= √2 - √6
4
3
7.3­7.4_Sum_Diff_Double_Half
March 21, 2012
Finding exact sine function values:
sin(A + B) = sin A cos B + cos A sin B
sin(A ­ B) = sin A cos B ­ cos A sin B
FIND:
sin(­150) = sin (300 + sin ­450)
= sin(300)cos(­450) + cos(300)sin(­450)
= (1)(√2) + √3(-√2)
2 2
2
2
= √2 - √6
4
4
7.3­7.4_Sum_Diff_Double_Half
March 21, 2012
Finding exact sine function values:
FIND:
5
7.3­7.4_Sum_Diff_Double_Half
March 21, 2012
Finding exact sine function values:
FIND:
6
7.3­7.4_Sum_Diff_Double_Half
March 21, 2012
Cofunction Identities
cos(900 ­ θ) = sin θ sec(900 ­ θ) = csc θ sin(900 ­ θ) = cos θ csc(900 ­ θ) =sec θ tan(900 ­ θ) = cot θ cot(900 ­ θ) = tan θ note: other identities can be obtained by replacing 900 with π/2
Find an angle θ that satisfies each of the following:
a) sec θ = csc 620
b) tan θ = cot -540
c) cos θ = sin 7π
6
7
7.3­7.4_Sum_Diff_Double_Half
March 21, 2012
Suppose that A and B are angles in standard position with:
Find:
a)
Sin (A ­ B)
b) tan(A ­ B)
c) the quadrant of A ­ B
8
7.3­7.4_Sum_Diff_Double_Half
March 21, 2012
9
7.3­7.4_Sum_Diff_Double_Half
March 21, 2012
7.4: Double Angle Identities
Cosine:
Sine:
tangent:
Given sin θ = 8 and cos θ < 0 find :
17
sin 2θ
cos2θ
tan 2θ
10
7.3­7.4_Sum_Diff_Double_Half
March 21, 2012
Find the values of the six trig functions of θ
if cos 2θ= -12/13 and 1800<θ<2700
11