multiple angle formulas for sine and tangent
DESCRIPTION
Multiple Angle Formulas for Sine and Tangent. Big formula day Again, use them, don’t memorize them (6.3, 6.4) (2). POD #1. Simplify. Rationalize the denominator. POD #2. Use one of yesterday’s formulas to verify the reduction formula. What is the relationship between θ and π /2 – θ ?. - PowerPoint PPT PresentationTRANSCRIPT
Multiple Angle Formulas for Sine and Tangent
Big formula day
Again, use them, don’t memorize them
(6.3, 6.4) (2)
POD #2
Use one of yesterday’s formulas to verify the reduction formula.
What is the relationship between θ and π/2 – θ ?
sin
2cos
POD #2
Verify the reduction formula.
How does this illustrate the relationship between cosθ and sinθ? How about their graphs?
How else could we write the cosine side?
sinsin
sin1sin0cos
sinsin2
sincos2
cos
sin2
cos
Cofunction formulas
Now, we move on to the trig formulas of composite angles of sine and tangent. We will build off of the cosine formulas we’ve already explored.
To begin with, however, we need to consider Cofunction formulas; these describe the relationship between the trig functions of complementary angles (get it?)
Cofunction formulas
Sine and cosine, tangent and cotangent, and secant and cosecant are related by their values for an angle θ, and its complement (π/2 – θ).
uu
uu
cos2
sin
sin2
cos
tan2
u
cot u
cot2
u
tanu
sec2
u
cscu
csc2
u
secu
Cofunction formulas
Although we can prove this in fairly short order, we can also simply look at a right triangle to see why this is so.
Addition Formula for Sine
Let’s derive sin(u + v) using cos(u + v). Use two cofunction formulas, too.
vuvu
vuvu
vu
vuvu
sincoscossin
sin2
sin)cos(2
cos
2cos
)(2
cos)sin(
uu
uu
sin2
cos
cos2
sin
The Rest
We can use this method to derive the rest of the formulas, but we won’t. Instead they are simply presented. You do not need to memorize them. You do need to use them.
Addition and Subtraction Formulas
For sine and tangent.
sin(u v) sinucosv cosusinv
sin(u v) sinucosv cosusinv
tan(u v) tanu tanv1 tanutanv
tan(u v) tanu tanv
1 tanutanv
Half-Angle Identities/ Formulas
For sine and tangent
u
uu
uu
u
uu
uu
cos1
cos1
2tan
2
cos1
2sin
2cos1
2cos1tan
2
2cos1sin
2
2
So, use them
1. If sin α = 4/5, and α is acute, find the exact values of sin 2α and cos 2α.
Do we care what α is?
So, use them
1. If sin α = 4/5, and α is acute, find the exact values of sin 2α and cos 2α.
To start with, we’ll need cos α. Think Pythagoras. How many ways could we find that value?
So, use them
1. If sin α = 4/5, and α is acute, find the exact values of sin 2α and cos 2α.
To start with in one approach, we’ll need cos α. Think Pythagoras.
cos α = 3/5
So, use them
1. If sin α = 4/5, and α is acute, find the exact values of sin 2α and cos 2α.
cos α = 3/5
sin 2α = 2(sin α)(cos α) = 2(4/5)(3/5)= 24/25
cos 2α = cos2 α - sin2 α = (3/5)2 – (4/5)2 = -7/25
How could we check our answers?
So, use them
2. Verify the identity. Take a deep breath– look at the formulas you have on both handouts. There are two that will work very well.
)4cos1(8
1cossin 22 xxx
So, use them
2. Verify the identity.
)4cos1(8
1
8
4cos1
42
4cos124
2
4cos11
4
2cos1
2
2cos1
2
2cos1
)4cos1(8
1cossin
2
22
xx
x
xx
xx
xxx
So, use them
2. Verify the identity. Alternate solution.
)4cos1(8
1
8
4cos1
42
4cos14
2sin
4
2cos1
2
2cos1
2
2cos1
)4cos1(8
1cossin
22
22
xx
x
xx
xx
xxx
So, use it– calculus preview
3. If f(x) = sin x, and h ≠ 0, show
hx
hx
h
xfhxf sinhcos
1coshsin
)()(