Core 4 compound angles?

Question

How do you do this?

TanP=7/24
TanQ=1

Work out- 

SIn(P+Q)
Cos(P+Q)
Tan(P-Q)

Answer 1

if tanP = 7 / 24 then sinp = 7 / 25 and cosp = 24 / 25

if tanQ = 1 then Q = pi / 4

Therefore sinQ = 1 / √2 and cosQ = 1 / √2

Recall : Trigonometry compound angles formulas sin(A + B) = sinAcosB + cosAsinB

sin(P + Q) = sinPcosQ + cosPsinQ = 7 / 25(1 / √2) + 24 / 25(1 / √2)

                                                                = 1 / √2(7 / 25 + 24 / 25)

                                                                = 1 / √2(31 / 25).

Answer 2

cos(P + Q)

if tanP = 7 / 24 then sinp = 7 / 25 and cosp = 24 / 25

if tanQ = 1 then Q = pi / 4

Therefore sinQ = 1 / √2 and cosQ = 1 / √2

Recall : Trigonometry compound angles formulas cos(A + B) = cosAcosB - sinAsinB

cos(P + Q) = cossPcosQ - sinPsinQ = 24 / 25(1 / √2) - 7 / 25(1 / √2)

                                                                = 1 / √2(24 / 25 - 7 / 25)

                                                                = 1 / √2(17 / 25).

 

Answer 3

TanP = 7 / 24 and TanQ = 1 then

Recall : Trigonometry compound angles Tan(A - B) = (TanA - TanB) / (1 + TanATanB)

Tan(P - Q) = (TanP - TanQ ) / (1 + TanPTanQ)

                   = (7 / 24 - 1) / ( 1 + 7 / 24(1))

                   =(-17 / 24) / (31 / 24)

                   =-17 / 31.

Answer 4

Given tanp = 7/24 and tanq = 1

From trigonometric identities: tan(theta) = opposite side/adjecent side

Comparing with tanp

tanp = 7/24 = opposite side/ adjacent side

(Hypo)^2 = opp side^2 + adj side^2

Hypo^2 = 7^2 + 24^2

hypo = 25

Sinp = opp/hypo = 7/25 , Cosp = adj/hypo = 24/25 ------------------(1)

Now, tanq = 1 = opposite side/adjacent side

tanq = 1/1 = opposite side/adjacent side

hypo^2 = opp^2 + adj^2

hypo^2 = 1 +1

hypo = sqrt2

sinq = 1/sqrt2 and cosq = 1/sqrt2-------------(2)

1. sin(p+q) 

Recall compound angle formulas: sin(p+q) = sinp*cosq + cosp*sinq  

Substitute sinp,sinq,cosp and cosq values from (1) and (2)

Sin(p+q) = 7/25*1/sqrt2 + 24/25*1/sqrt2

Sin(p+q) = (7+24)/(25*sqrt2)

Sin(p+q) = 31/(25*sqrt2)

2. cos(p+q)

Recall formula: cos(p+q) = cosp*cosq - sinp*sinq

Substitute cosp,cosq ,sinp,sinq values from (1) and(2)

Cos(p+q) = 24/25*1/sqrt2 - 7/2581/sqrt2

Cos(p+q) = 24-7/(258sqrt2)

Cos(p+q) = 17/(25*sqrt2)

3. tan(p-q)

Recall formula : tan(p-q) = (tanp-tanq)/(1+tanp*tanq)

Tan(p-q) = (7/24 - 1)/(1+(7/24)*1)

Tan(p-q) = (7-24)/24)/(1+7/24)

Tan(p-q) = (-17/24)/((24+7)/24

Tan(p-q) = -17/31