Trigonometry help?

Question

4. Find cos[(3π/4) - (π/3)].

5. Solve the following trigonometric equation such that 0 ≤ x ≤ 2π .

[tanx + tan(π/3)] / [1 - (tanx) tan(π/3)] = √3

6. Prove the co-function identity: tan[ x + (π/2)] = - cot θ.

Answer 1

4).

To find cos (3π/4 - π/3),

Apply property : cos (x - y) = cos x cos y + sin x sin y.

cos (3π/4 - π/3) = cos 3π/4 cos π/3 + sin 3π/4 sin π/3

Substitute the values cos 3π/4 = - 1/√2, cos π/3 = 1/2, sin 3π/4 = 1/√2, and sin π/3 = √3/2.

= (- 1/√2)(1/2) + (1/√2)(√3/2)                                   

= (- 1/2√2) + (√3/2√2)

= ( √3 - 1 )/2√2.

Therefore, cos (3π/4 - π/3) = √3 - 1 )/2√2.

5).

[tan x + tan (π/3)] / [1 - (tan x) tan (π/3)] = √3.

Apply property : tan (x + y) = [tan x + tan y] / [1 - (tan x tan y) ].

tan (x + π/3) = √3

tan (x + π/3) = tan (π/3)

x + π/3 = π/3

x = 0.

Therefore, there is no solution exists in the interval [0, 2π].

general solution x + π/3 = nπ + π/3.

⇒ x = nπ, where n ia an integer.

Comments

The trigonometric equation is [tan x + tan (π/3)] / [1 - (tan x) tan (π/3)] = √3, where 0 ≤ x ≤ 2π.

Apply sum formula : tan(u + v) = [ tan u + tan v ] / [ 1 - tan u tan v ].

tan[x + π/3] = √3

tan[x + π/3] = tan(π/3)

x + π/3 = π/3

x = 0.

To check the solution, substitute the value of x = 0 in the original equation.

[tan x + tan (π/3)] / [1 - (tan x) tan (π/3)] = √3

[tan(0) + tan (π/3)] / [1 - tan(0) tan (π/3)] = √3

[0 +√3 ] / [1 - (0)(√3) ] = √3

√3 = √3.

The above statement is true, the value of x = 0 is the solution of original equation.

Answer 2

it is given that:

cot(x) = tan(90 - x)

this assumes both your angles are in quadrant 1.

you have angle x and you have angle 90 - x.

the equivalent angles in the second quadrant are (180 - x) and (180 - (90 - x)).

the reference angle for (180 - x) is equal to x.

the reference angle for (180 - (90 - x)) is equal to 90 - x).

since (180 - x) is in quadrant 2, then tan(180 - x) = -tan(x) and cot(180 - x) = -cot(x).

since 180 - (90 - x) is in quadrant 2, then tan(180 - (90 - x)) = -tan((90 - x) and cot(180 - (90 - x) = -cot(90 - x).

we'll start with:

cot(180 - x) = tan(180 - (90 - x))

this is true because the reference angle for (180 - x) is x and the reference angle for (180 - (90 - x)) is (90 - x) and we already know that cot(x) = tan(90 - x).

we also know that cot(180 - x) is equal to -cot(x) because the cotangent function in the second quadrant is the negative of the cotangent function in the first quadrant for the angles that have the same reference angle. since the reference angle for (180 - x) is x, this rule applies.

our equation therefore becomes:

-cot(x) = tan(180 - (90 - x))

we can simplify tan(180 - (90 - x) to be equivalent to tan (90 + x).

this completes the proof.

http://www.algebra.com/algebra/homework/Trigonometry-basics.faq.question.870083.html.