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.

M IB

M2I EN2

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

Double Angle Formulas

For sine and tangent

sin2u2sinucosu

tan2u 2tanu

1 tan2 u

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

)()(

So, use it– calculus preview

3. If f(x) = sin x, and h ≠ 0, show

What is the advantage to having the expression in this form?

hx

hx

h

x

h

xxh

xxxh

xhx

h

xfhxf

sinhcos

1coshsin

sinhcossincoshsin

sinsinhcoscoshsin

sin)sin()()(