Advertisements
Advertisements
Question
If cos θ = `1/2 ("a" + 1/"a")`, show that cos 3θ = `1/2 ("a"^3 + 1/"a"^3)`
Advertisements
Solution
cos θ = `1/2 ("a" + 1/"a")`
cos 3θ = 4 cos3θ – 3 cos θ
= `4[1/2("a" + 1/"a")]^3 - 3[1/2("a" + 1/"a")]`
= `4 xx 1/8("a" + 1/"a")^3 - 3/2("a" + 1/"a")`
= `1/2("a" + 1/"a")^3 - 3/2("a" + 1/"a")`
= `1/2["a"^3 + 3"a"^2(1/"a") + 3"a"(1/"a")^2 + 1/"a"^3] - 3/2(a" + 1/"a")`
= `1/2["a"^3 + 3"a" + 3/"a" + 1/"a"^3] - 3/2"a" - 3/(2"a")`
= `1/2 "a"^3 + 3/2"a" + 3/(2"a") + 1/(2"a"^3) - 3/2"a" - 3/(2"a")`
cos 3θ = `1/2"a"^3 + 1/(2"a"^3)`
= `1/2("a"^3 + 1/"a"^3)`
APPEARS IN
RELATED QUESTIONS
Find the values of tan(1050°)
`(5/7, (2sqrt(6))/7)` is a point on the terminal side of an angle θ in standard position. Determine the six trigonometric function values of angle θ
Find the value of the trigonometric functions for the following:
cos θ = `- 2/3`, θ lies in the IV quadrant
Show that `sin^2 pi/18 + sin^2 pi/9 + sin^2 (7pi)/18 + sin^2 (4pi)/9` = 2
If sin A = `3/5` and cos B = `9/41 0 < "A" < pi/2, 0 < "B" < pi/2`, find the value of sin(A + B)
Find the value of sin105°.
Find a quadratic equation whose roots are sin 15° and cos 15°
Show that tan 75° + cot 75° = 4
Prove that sin2(A + B) – sin2(A – B) = sin2A sin2B
Prove that (1 + tan 1°)(1 + tan 2°)(1 + tan 3°) ..... (1 + tan 44°) is a multiple of 4
Prove that sin x + sin 2x + sin 3x = sin 2x (1 + 2 cos x)
Prove that 1 + cos 2x + cos 4x + cos 6x = 4 cos x cos 2x cos 3x
Prove that `(sin(4"A" - 2"B") + sin(4"B" - 2"A"))/(cos(4"A" - 2"B") + cos(4"B" - 2"A"))` = tan(A + B)
If A + B + C = 180◦, prove that sin 2A + sin 2B + sin 2C = 4 sin A sin B sin C
If A + B + C = 180°, prove that `tan "A"/2 tan "B"/2 + tan "B"/2 tan "C"/2 + tan "C"/2 tan "A"/2` = 1
If A + B + C = 2s, then prove that sin(s – A) sin(s – B)+ sin s sin(s – C) = sin A sin B
If A + B + C = `pi/2`, prove the following cos 2A + cos 2B + cos 2C = 1 + 4 sin A sin B sin C
If ∆ABC is a right triangle and if ∠A = `pi/2` then prove that sin2 B + sin2 C = 1
