19.03.25 Trigonometric Identities

Half Angle

100

Find the value of sin75°.

sqrt(2+sqrt3)/2

200

Find the value of

tan((7π)/12)

-2-sqrt(3)

300

Solve

2sin^2(x/2)+cosx=1+sinx 

on the interval of

[0,π]

0, pi

400

Solve

2tan(x/2)+2cos(x)tan(x/2) = 1

 on the interval [0,2π].

π/6 and 5π/6

500

Find

tan(x/2)

 when

tanx=-1/2 

and x is in the second quadrant. 

2 - sqrt(5)

Double Angle

100

Solve

sin2x=cosx

 on the interval [0, 2π].

x=π/6 and 5π/6

200

Solve

1-sin^2x-cos2x=3/4

π/3 and 5π/3

300

Find sin2π when sinx=-5/13 and "x IS GREATER THAN 3π/2 AND LESS THAN 2π."

-120/169

400

Solve cos2x=cosx on the interval [0,2π].

0 and π and 4π/3 and 4π/5

500

Find cos2x when tanx=-1/2 and "x IS GREATER THAN π/2 x AND LESS THAN π."

3/5

Pythagorean Identities

100

Simplify the expression to a single trigonometric function 

(1 – cos^2x)(cscx)

sinx

200

Simplify this expression into an expression whose only trig function is sin: 

sin^4(x) - cos^4(x)

2sin^2(x) - 1

300

Simplify this complex fraction into a single trigonometric function: 

[(sinx/cosx) + (cosx/sinx)]/[(1)/(sinx)]

1/cosx

400

Simplify: 

[(1 - cos^2(x))]/[(sec^2(x) - 1)

cos^2(x)

500

If… csc(x) = 5/3 and tan(x) = 3/4, find the values of the remaining trigonometric functions using a Pythagorean Identity.

cos(x) = +4/5

Sum and Difference

100

Find the exact value of the following expression: sin147°cos78°+cos147°sin78°

-(√2)/2

200

Find the exact value of cos(11π/12) using the sum and difference formula

(-√2 - √6)/4

300

If cosƟ = ¼ and 3π/2 ≤ Ɵ ≤ 2π, find… Sin(Ɵ – π/6)

(-3√5-1)/8

400

Establish the identity csc(Ɵ – π/2) = -secƟ

1/(sin(Ɵ - π/2)) = 1/(sinƟcosπ/2 - cosƟsinπ/2) = 1/(sinƟ(0) - cosƟ(1)) = 1/(-cosƟ) = -secƟ = -secƟ QED

500

Establish the identity tanx – tany = (sin(x-y))/((cosx)(cosy))

(sin(x-y))/((cosx)(cosy)) = (sinx(cosy) - cosx(siny))/((cosx)(cosy)) = (sinx (cosy))/((cosx)(cosy)) – (cosx(siny))/((cosx)(cosy)) = tanx(1) – 1(tany) = tanx – tany = tanx – tany QED

More Identities

100

What is the equation for sin(x/2)?

±√(1-cosx)/2

200

Simplify the equation cos(2x) in terms of sin.

1-(2sin^2)x

300

Find the value of cos(sin^-1(-5/13))

12/13

400

prove that (sin^4x - cos^4x)/(sin^2x - cos^2x) = 1

(sin^4x - cos^4x)/(sin^2x - cos^2x) = ((sin^2x)^2 - (cos^2x)^2)) / (sin^2x - cos^2x) = (sin^2x+cos^2x)(sin^2x - cos^2x) / (sin^2x - cos^2x) = sin^2x+cos^2x = 1 = 1 QED

500

Prove that cosx/(1+sinx) + (1+sinx)/cosx = 2secx ?

cosx/(1+sinx) + (1+sinx)/cosx = (cos^2)x/(1+sinx)cosx + (1+sinx)^2/(1+sinx)cosx = ((cos^2)x + (1+sinx)^2) / (1+sinx)cosx = ((cos^2)x + 1 + 2sinx + sin^2x)/(1+sinx)cosx = ((cos^2)x ?+ sin^2x + 1 + 2sinx) /(1+sinx)cosx = 2 + 2sinx/(1+sinx)cosx = 2(1 + sinx)/(1+sinx)cosx = 2/cosx = (2)(1/cosx) = 2secx = 2secx QED