Advanced Identities: Your Keys to Pre-Calc Success

Question 1

1) What to do if you need to find out a sine/cosine/tangent 15 degree angle? What are the 6 equations to do so?
2) Praxis Example: find the sine of 135°!
3) What is the common denominator of the unit circle?
4) What to do if you need to find out the value of sin(π/12)?

Answer 1

1) Use the sum or difference of known angles. So 45-30 degrees.
sin(a + b) = sin a · cos b + cos a · sin b
sin(a – b) = sin a · cos b – cos a · sin b

cos(a + b) = cos a · cos b – sin a · sin b
cos(a – b) = cos a · cos b + sin a · sin b

tan(a+b)= tan a + tan b
____________
1 - tan a * tan b

tan(a-b)= tan a - tan b
____________
1 + tan a * tan b

2)
- Rewrite. P.e. 135° = 90° + 45°
- Use sum formula: sin 90° cos 45° + cos 90° sin 45°
- Unit circle to transform to: 1SR2/2+0SR2/2 -> mind sin table and that the cartesian above is (1,0)
- Simplify to SR2/2

3) 12
4) Same as above: Rewrite (3π/12-2π/12), use formula, transform and simplify.

Question 2

Find the sum or difference of two angles…

Find the exact value of cos(A + B), given that
cos A = –3/5, with A in quadrant II of the coordinate plane, and sin B = –7/25, with B in quadrant III.

Answer 2

1. Choose a formula and substitute given information.
   cos(A + B) = (–3/5) · cos B – sin A · (–7/25)
2. Draw the 4 quadrants and the right triangles in the quadrants (A in two, B in 3). Also mark the length of the two sides you know. Then use Pythagoras to find third. Picture on page 321.
3. Use the definition of cosine and sine.
   You get cos B = –24/25 and sin A = 4/5.
4. Put into the formula and simplify.
   cos(A + B) = (–3/5) · (–24/25) – (4/5) · (–7/25)
   cos(A + B) = 4/5.

Question 3

Name the double-angle formulas and derivations for sine, cosine, tangent!

Answer 3

if … sin(a + b) = sin a · cos b + cos a · sin b
then … sin(θ+θ) = sin θ · cos θ + cos θ · sin θ
thus … sin(2θ) = 2 sin θ · cos θ (expression above 2x)

if … cos(a + b) = cos a · cos b – sin a · sin b
then … cos(θ+θ) = cos θ · cos θ - sin θ · sin θ
thus … cos(2θ) = cos2 θ - sin2 θ

also possible: cos 2x = 2 cos² x – 1
cos 2x = 1 – 2 sin² x

if … tan(a+b) = tan a + tan b
__________
1 - tan a * tan b

then … tan(θ+θ)= tan θ + tan θ
____________
1 - tan θ * tan θ

thus … tan(2θ)= 2 tan θ
________
1 - tan**2 θ

Question 4

Name the half-angle formulas for sine, cosine, and tangent.

Answer 4

sin x/2 = +- √1-cos x
____ # nom and denom under SR
2

cos x/2 = +- √1+cos x
____ # nom and denom under SR
2

tan x/2 = 1 - cos x sin x
______ = _____
sin x 1 + cos x

Question 5

Name the three product-to-sum formulas sine · cosine, cosine · cosine, and sine · sine!

Answer 5

sine · cosine
-> sin a cos b = 1/2[sin(a+b) + sin(a-b)]

cosine · cosine
-> cos a cos b = 1/2[cos(a+b) + cos(a-b)]

sine · sine
-> sin a sin b = 1/2[cos(a-b) - cos(a+b)]

Question 6

Name the Sum/Difference-to-Product identities!

Answer 6

sin x + sin y = 2sin(x+y)cos(x-y)
__ __ #den also in parenth
2 2

sin x - sin y = 2cos(x+y)sin(x-y)
__ __ #den also in parenth
2 2

cos x + cos y = 2cos(x+y)cos(x-y)
__ __ #den also in parenth
2 2

Question 7

Name the Power-Reducing Formulas to eliminate exponents

Answer 7

sin**2 x = 1-cos2x
_____
2

cos**2 x = 1+cos2x
_____
2

tan**2 x = 1-cos2x
_____
1+cos2x