5. Identities and Formulas 5.1 5.2 5.3 5.4 5.5

Proving Identities Sum and Difference Formulas Double-Angle Formulas Half-Angle Formulas Additional Identities

1

5.1 Proving Identities Guidelines for Proving Identities (my short ver.) 1) 2) 3) 4) 5)

Work complicated side first Use the basic identities Look for algebraic operations Change everything to sines and cosines Work on one side, keep an eye on the other side.

2

5.1 Proving Identities Prove. (1) secθ cotθ = cscθ [4] (2) sin x (sec x + csc x) = tan x + 1

[10]

sin 4 t − cos4 t 2 2 = sec t − csc t (3) 2 2 sin t cos t 2 cos θ (4) 1 − sin θ = [20] 1 + sin θ

[18]

(5)

sec B − cos B = tan B sin B

[30]

(6)

1 1 + = 2 sec 2 x 1 − sin x 1 + sin x

[36] 3

5.1 Proving Identities Prove. sin t 1 − cos t = 1 + cos t sin t

(7)

[40]

(8) Prove the following statement is not an identity by find a counterexample. [68, 70] •

Sinθ + cosθ = 1

•

tan2θ + cot2θ = 1 4

5.2 Sum and Difference Formulas 1) 2) 3) 4)

Cosine of sum and of difference Sine of sum and of difference Tangent of sum and of difference Demonstrations

5

5.2 Sum and Difference Formulas 1) Cosine of sum and of difference cos(A + B) = cosA cosB – sinA sinB cos(A – B) = cosA cosB + sinA sinB • •

The 2nd formula follow from the 1st, A – B = A + (–B) The prove of 1st formula is in the book.

6

5.2 Sum and Difference Formulas 2) Sine of sum and of difference sin(A + B) = sinA cosB + cosA sinB sin(A – B) = sinA cosB – cosA sinB • •

The 2nd formula follow from the 1st, A – B = A + (–B) The 1st formula using sin(A) = cos(90° – A)

7

5.2 Sum and Difference Formulas 3) Sine of sum and of difference

tan A + tan B tan( A + B ) = 1 − tan A tan B tan A − tan B tan( A − B ) = 1 + tan A tan B • •

Using tan = sin/cos. Divide both numerator and denominator by cosAcosB

8

5.2 Sum and Difference Formulas 4) Problems (1) Find the exact value of sin75°

[2]

(2) Prove: cos(90 + θ) = –sin(θ)

[16]

Ans.

2+ 6 4

(3) Write the expression as a single trig function: [24] sin(8x)cos(x) – cos(8x)sin(x) Ans. sin(7x) (4) Graph one cycle by writing the function as a single trig function: y = 2(sin x cos π3 – cos x sin π3 ) [34] y = 2 sin( x − π3 ) 9

5.2 Sum and Difference Formulas 4) Problems (5) Let cos A = − 135 with A ∈ QII and sin B = 53 with B ∈ QI. Find sin(A – B), cos(A – B) and tan(A – B). In what quadrant does A – B terminate? [36] sin( A − B ) =

63 65

63 , cos( A − B ) = 16 , tan( A − B ) = 65 16 ; A − B ∈ QI

(6) Find exact value of cos(76π )

[6]

Ans.

(7) If tan(A + B) = 2, and tanB = 13 , find tan A. [40]

2− 6 4

Ans. 1

10

5.3 Double-Angle Formula 1) 2) 3) 4)

Double-Angle formula for sine Double-Angle formula for cosine Double-Angle formula for tangent Problems

11

5.3 Double-Angle Formula 1) Double-angle formula for sine sin(2A) = 2sinA cosA •

In the sum formula, let B = A, we obtain above: sin(A + B) = sin(A)cos(B) + cos(A)sin(B).

12

5.3 Double-Angle Formula 2) Double-angle formula for cosine cos(2A) = cos2A – sin2A = 2cos2A – 1 = 1 – 2sin2A • •

first form second form third form

In the sum formula, let B = A, we obtain first form: cos(A + B) = cos(A)cos(B) – sin(A)sin(B). Using cos2A + sin2A = 1, we obtain 2nd and 3rd form.

13

5.3 Double-Angle Formula 3) Double-angle formula for tangent 2 tan A tan(2A) = 1 − tan 2 A

•

In the sum formula, let B = A, we obtain the above: tan A + tan B tan( A + B ) = 1 − tan A tan B

14

5.3 Double-Angle Formula 4) Problems (1) If cos x =

1 10

Ans. −

with x∈QIV, find sin(2x) [6]

(2) Simplify: sin(π8 )cos(π8 ). [34] (3) If sinA= − 53 with A∈QIII, find cos(2A) [2]

Ans.

3 5

2 4

7 25

Ans.

(4) Graph the function from x = 0 to x = 2π: [18] y = 2 – 4 sin2x

15

5.3 Double-Angle Formula 4) Problems (5) Simplify: 2cos2105° – 1

Ans. −

[32]

(6) Use exact value to show that sin90° = 2sin45°cos45° is true. (7) If tanA = – 3 , find tan(2A)

tan 38π (8) Simplify: 1 − tan 2 38π

[36]

[28]

3 2

[26]

Ans.

3

Ans. –0.5

16

5.3 Double-Angle Formula 4) Problems (9) prove

1 − cos(2θ ) sin θ = 2 2

[40]

θ

sin(2θ ) in terms of just x. (10) If x = 4sinθ, write expression − 2 4 [62] 1 2

sin −1 ( 4x ) − 321 x 16 − x 2

17

5.4 Half-Angle Formulas 1) 2) 3) 4)

Half-Angle formula for sine Half-Angle formula for cosine Half-Angle formula for tangent Problems

18

5.4 Half-Angle Formulas 1) Double-angle formula for sine

sin • •

A 2

1 − cos A =± 2

Obtained from half angle formula for cosine, 3rd form. Need to decide the sign by determine which quadrant does half angle A/2 belong.

19

5.4 Half-Angle Formulas 2) Double-angle formula for cosine

cos • •

A 2

1 + cos A =± 2

Obtained from half angle formula for cosine, 3rd form. Need to decide the sign by determine which quadrant does half angle A/2 belong.

20

5.4 Half-Angle Formulas 3) Double-angle formula for tangent 1st

formula

2nd formula • •

1 − cos A tan = sin A sin A A tan 2 = 1 + cos A A 2

Obtained using double-angle formula for sine and cosine. Need to decide the sign.

21

5.4 Half-Angle Formulas 4) Problems (1) If 90° < A < 180°, then A/2 terminate in which quadrant? [2] Ans. quadrant I (2) If 270° < A < 360°, then A/2 terminate in which quadrant? [4] Ans. quadrant II (3) If sinB = − (i) csc B2 6 3+ 8

1 3

with B in QIII, find: (ii) sec B2

−

6 3− 8

[18, 20, 22] (iii) cot B2

− 3+1 8 22

5.4 Half-Angle Formulas 4) Problems (4) If sinA = 45 with A in QII, sinB = 53 with B in QI, find 1 (i) cos 2A [24] (ii) sin B2 [30] 5

(5) Graph from 0 to 4π: (i) y = 6cos2 2x

1 10

[36, 38] (ii) 2sin2 2x

(6) Use half-angle formula to find the exact value: [40, 42] (i) tan15° (ii) cos75°

2− 3

2− 3 2 23

5.4 Half-Angle Formulas 4) Problems Prove the following identities: (7) 2 cos2 θ2

sin 2 θ = 1 − cos θ

(8) tan B2 =

sec B sec B csc B + csc B

24

5.5 Additional Identities 1) 2) 3) 4)

Identities and formulas involving inverse function Product to Sum formulas Sum to Product formulas Problems

25

5.5 Additional Identities 1) Identities and formulas involving inverse function Evaluate the given expression without using calculator. 2 (1) cos(arcsin 53 – arctan 2) [2] 5

(2) cos(2sin–1 13 )

[8]

7 9

Write each expression as an equivalent expression involving only x (assume x is positive) [12, 18] (3) cos(tan–12x ) (d) sin(sec–1 x3+1 ) 2 x 2 +4

x 2 + 2 x −8 x +1 26

5.5 Additional Identities 2) Product to Sum formulas 1 sin A cos B = [sin( A + B ) + sin( A − B ) ] 2 1 cos A sin B = [sin( A + B ) − sin( A − B ) ] 2 1 cos A cos B = [cos( A + B ) + cos( A − B ) ] 2 1 sin A sin B = [cos( A − B ) − cos( A + B ) ] 2

•

(1) (2) (3) (4)

Expand the right side using sum/difference formula

27

5.5 Additional Identities 2) Product to Sum Formula (1) Verify formula (1) for A = 120° and B = 30°.

[20]

(2) Rewrite each expression as a sum or difference, simplify (i) 10sin(5x)sin(3x) [22] 5[cos(2x) – cos(8x)] (ii) cos(2x)sin(8x)

[24]

[sin(10x) + sin(6x)]/2

(iii) cos(3π)sin(π)

[28]

[sin(4π) + sin(2π)]/2 = 0

28

5.5 Additional Identities 3) Sum to Product formulas sin α + sin β = 2 sin

α +β

sin α − sin β = 2 cos

α +β

2

cos α + cos β = 2 cos cos α − cos β = − 2 sin

•

cos sin

2

α +β 2

(5)

2

α −β

cos

α +β 2

α −β

sin

(6)

2

α −β 2

α −β 2

(7) (8)

Obtained from Sum to Product formula by substituting: α = A + B, β = A – B (thus, A = (α + β)/2, B = (α – β)/2 29

5.5 Additional Identities 3) Sum to Product formulas (1) Verify formula (8) for A = 90° and B = 30°.

[30]

(2) Rewrite each expression as a sum or difference, simplify (i) cos(5x) – cos(3x) [32] –2sin(4x)sin(x) (ii) sin75° – sin15°

[34]

(iii) cos(12π ) + cos( 712π )

[36]

(3) Verify the identity: tan( 4 x ) =

2 sin 45 o sin 30 o = 2 sin

π 3

sin

π 4

cos( 3 x ) − cos( 5 x ) sin( 3 x ) − sin( 5 x )

=

1 2

1 2

[40] 30