# trigonometry math Questions and Answers

• trigonometry math
We know that the expression for cos ( x + y) is: cos (x + y) = cos x * cos y - sin x* sin y Now substituting x for both x and y we get cos ( x + x) = cos x * cos x - sin x * sin x => cos 2x =...

• trigonometry math
We have to prove [sin 3a + sin a + sin 5a]/ [cos 3a+cos a+cos 5a] = tan 3a We start with [sin 3a + sin a + sin 5a]/ [cos 3a+cos a+cos 5a] We use the formula: sin A + sin B = 2 sin [ (A + B) / 2 ]...

• trigonometry math
We have to verify sin 25 + sin 35 = cos 5 sin 25 + sin 35 => 2* sin (25 + 35)/2 * cos 10/2 => 2* sin (60)/2 * cos 5 => 2* sin 30 * cos 5 sin 30 = 1/2 => 2 * (1/2) cos 5 => cos 5 This...

• trigonometry math
We have to prove that: (cos x)^2 - (sin x)^2 = 2*(cos x)^2 - 1 Starting with the left hand side: We use the relation (cos x)^2 + (sin x)^2 = 1 or (sin x)^2 = 1 - (cos x)^2 => (cos x)^2 - (sin...

• trigonometry math
First, let us rewrite the cube of sine and cosine in the left-hand side of the above equation as a product of a square of the sine/cosine and the sine/cosine: (sinx)^2*sinx*sin(3x) +...

• trigonometry math
We need to simplify the expression (cos (x-y))^2 + (cos (x+y))^2 - cos 2x *cos 2y (cos (x-y))^2 + (cos (x+y))^2 - cos 2x *cos 2y => (cos x * cos y + sin x * sin y)^2 + (cos x * cos y - sin x *...

• trigonometry math
We have to find the result of sin(pi/3) + sin(2pi/3) + sin(3pi/3) + sin(4pi/3) We use sin (pi/3) = (sqrt 3)/2 sin (2*pi/3) = sin (pi - pi/3) = sin (pi/3) = (sqrt 3)/2 sin (3*pi/3) = sin (pi) = 0...

• trigonometry math
We have to find the zeros of f(x) = sin x + cos x f(x) = sin x + cos x = 0 => sin x = -cos x => tan x = -1 x = arc tan (-1) In the interval (0, pi) , x = -pi/4 + pi = 3*pi/4 The required...

• trigonometry math
We have sin (a+b) = (sin a * cos b + cos a * sin b) and cos(a + b) = (cos a * cos b – sin a * sin b) Also tan (a + b) = [sin (a+b)] / [cos (a+b)] => [sin a * cos b + cos a * sin b] / [cos a *...

• trigonometry math
We have sin a + cos a = 1/3 Use the fact that sin 2a = 2*(sin a)(cos a) sin a + cos a = 1/3 square the right and left hand sides (sin a + cos a)^2 = (1/3)^2 => (sin a)^2 + (cos a)^2 + 2*(sin...

• trigonometry math
We have to find x given that : tan^2 x - 4sec x + 5 = 0 (tan x)^2 - 4sec x + 5 = 0 => ( 1 - (cos x)^2)/( cos x)^2) - 4/ cos x + 5 = 0 let cos x = y => (1 - y^2)/y^2 - 4/y + 5 = 0 => (1 -...

• trigonometry math
sin (x + y) = sin x* cos y + cos x * sin y It is given that sin x + cos y = 1/4 and cos x + sin y = 1/2 sin x + cos y = 1/4 square the two sides => (sin x)^2 + (cos y)^2 + 2*sin x*cos y = 1/16...

• trigonometry math
Here, we use the relations that tan x = sin x / cos x. sin (a + b) = sin a * cos b + cos a * sin b cos ( a+ b) = cos a * cos b - sin a * sin b tan (a + b) = sin (a + b) / cos ( a + b) => [sin a...

• trigonometry math
The expression you have provided is true because sin 2x = 2*sin x*cos x and cos 2x = (cos x)^2 - (sin x)^2 Let start with (cos x + sin x)/(cos x - sin x) multiply the numerator and denominator by...

• trigonometry math
To find the distance from the vertex to the shoreline which will divide the lot into two equal parts, find the length of what would be the line dividing the lot. As we have 2 exisitng angles, find...

• trigonometry math
We have to solve 1 + cos x + cos 2x = 0 for x in the interval (0, pi) 1 + cos x + cos 2x = 0 => 1 + cos x + 2(cos x)^2 - 1 = 0 => cos x + 2(cos x)^2 = 0 => cos x( cos x + 2) = 0 => cos...

• trigonometry math
We have to solve cos 3x = cos x for x in the interval (0 , 2*pi) We know that cos 3x = 4(cos x)^3 - 3cos x cos 3x = cos x => 4(cos x)^3 - 3cos x = cos x let cos x = t => 4t^3 - 4t = 0 =>...

• trigonometry math
We are given that sin x = (1/2) and sin y =(1/3). We have to find tan ( x - y). sin x = (1/2) => tan x = (1/2)/ sqrt [1 - (1/2)^2] => (1/2)/ sqrt (3/4) => 1/ sqrt 3 sin y = (1/3) => tan...

• trigonometry math
We need the solutions of sin 12x + cos 6x = 0. sin 12x + cos 6x = 0 => 2*sin 6x*cos 6x + cos 6x = 0 let cos 6x = y => 2*sqrt(1 - y^2)*y + y = 0 => y(2*sqrt(1 - y^2) + 1) = 0 => y = 0...

• trigonometry math
We have to determine the trigonometric form of z = -2i The absolute value of z = -2i sqrt [ 0^2 + (-2)^2] => sqrt 4 => 2 and the the argument is -90 as the real component is zero. Therefore...

• trigonometry math
To find the value of sec (2x) if sin x = 3/5, cos x = 4/5, use the relation sin 2x = 2*sin x * cos x and (cos x)^2 = 1 - (sin x)^2 sin 2x = 2*sin x * cos x = 2*3/5 * 4/5 => sin 2x = 24/25 (cos...

• trigonometry math
We have that tan x = 6/11. tan x is equal to the opposite side/ adjacent side for the angle in a right triangle. Using the Pythagorean theorem we calculate the length of the hypotenuse as sqrt (...

• trigonometry math
We have to simplify: [﻿cos x*sin(2x) - 2 sin x]/(sin x)^2 *cos x [﻿cos x*sin(2x) - 2 sin x]/(sin x)^2 *cos x use sin 2x = 2*sin x * cos x => [2*(﻿cos x)^2 *sin x - 2 sin x]/(sin x)^2 *cos x...

• trigonometry math
The equation given is 99sin^2x-101=-sin^2x. The number of roots of the equation have to be determined. 99(sin x)^2 - 101 = -(sin x)^2 => 100(sin x)^2 = 101 => (sin x)^2 = 101/100 => sin x...

• trigonometry math
We have 2*sin x*sin 3x - cos 4x = 0 (sin a)(sin b) = (1/2)(cos (a - b) - cos (a + b)) => 2*(sin a)(sin b) = cos (a - b) - cos (a + b) 2*sin x*sin 3x - cos 4x = 0 => cos (x - 3x) - cos 4x -...

• trigonometry math
We have to prove that 1+ sin x = cos( 90 - x) + cot 45 Let's start with cos( 90 - x) + cot 45 cos (90 - x) - cot 45 cos 45 = sin 45 = 1/sqrt 2 , cos (90 - x) = sin x => sin x + cot 45 => sin...

• trigonometry math
We have to solve 3*cos(3x - 1) = 0 for x. 3*cos(3x - 1) = 0 => cos(3x - 1) = 0 => 3x - 1 = arc cos 0 => 3x - 1 = pi/2 + 2*n*pi and 3x - 1 = 3*pi/2 + 2*n*pi => 3x = pi/2 +1 + 2*n*pi and...

• trigonometry math
To prove the identity: AB^2+AC^2=2BM^2+2AM^2, we have to use the law of cosines. For the triangle AMB, c^2 = x^2 + y^2 - 2*x*y*cos (AMB)...(1) For the triangle AMC b^2 = x^2 + y^2 - 2*x*y*cos...

• trigonometry math
We have to prove that cos 2x * (1 + tan x*tan 2x) = 1 cos 2x = 1 - (sin x)^2 and tan 2x = 2*tan x/(1 - (tan x)^2) cos 2x * (1 + tan x*tan 2x) => ((cos x)^2 - (sin x)^2)*(1 + (tan x)*(2*tan x/(1...

• trigonometry math
We have to prove that (sin x)^2 * [ 1 + (cot x)^2] = 1 (sin x)^2 * [ 1 + (cot x)^2] => (sin x)^2 * [ 1 + (cos x)^2/ (sin x)^2] => (sin x)^2 * [ (sin x)^2 + (cos x)^2]/ (sin x)^2 => (sin...

• trigonometry math
We have to show that cos 3B + cos B = 2(cos 2B)* cos B We use the relations: cos 2B = 2*(cos B)^2 - 1 and cos 3B = 4(cos B)^3 - 3cos B We start with the left hand side cos 3B + cos B => 4(cos...

• trigonometry math
We have to solve 2 tan x - 1 = -tan (-x) Now we know that tan -x = - tan x 2 tan x - 1 = -tan (-x) => 2 tan x - 1 = tan (x) => tan x - 1 = 0 => tan x = 1 => x = arc tan 1 => x = 45...

• trigonometry math
The equation that has to be solved is (sin B)^2- 3 sin B + 2 =0 where 0 ≤ B ≤ 2π. (sin B)^2- 3*sin B +2 =0 => (sin B)^2- 2*sin B - sin B +2 =0 => (sin B)(sin B - 2 ) - 1( sin B - 2) =0 =>...

• trigonometry math
The equation to be solved is (cos x)^6 = 1 - (sin x)^6 (cos x)^6 = 1 - (sin x)^6 => (cos x)^6 + (sin x)^6 = 1 => (cos x)^2^3 + (sin x)^2^3 = 1 use a^3 + b^2 = (a + b)(a^2 - ab + b^2) =>...

• trigonometry math
We have to identify t if cos 2x + 2*(sin x )^2 + t = 0 cos 2x + 2*(sin x )^2 + t = 0 use cos 2x = 1 - 2*(sin x)^2 => 1 - 2*(sin x)^2 + 2*(sin x)^2 + t = 0 => 1 + t = 0 => t = -1 The...

• trigonometry math
We have to find x if (sin x)^2 = 1/4 (sin x)^2 = 1/4 => sin x = 1/2 and sin x = -1/2 x = arc sin (1/2) and x = arc sin (-1/2) arc sin (1/2) = 30 degrees and arc sin (-1/2) = -30 degrees. As the...

• trigonometry math
We have a right triangle with A = 90 degrees and we have to calculate cos B / sin C + cos C / sin B Let's take the sides opposite angle A as a, that opposite angle B as b and that opposite angle C...

• trigonometry math
We are given that 1 - 2(sin a)^2 = 2* (cos a)^2 + m. We have to find m. 1 - 2(sin a)^2 = 2* (cos a)^2 + m => 1 - 2(sin a)^2 - 2* (cos a)^2 = m => 1 - 2[(sin a)^2 + (cos a)^2] = m now use (sin...

• trigonometry math
We have to find the sum sin 135 + cos 150 Use the relations: sin (90 + x) = cos x and cos(180 - x) = -cos x sin 135 + cos 150 => sin (90 + 45) + cos (180 - 30) => cos 45 - cos 30 cos 45 =...

• trigonometry math
We have to find x for 100*sin x - 99*cos x = 0 100*sin x - 99*cos x = 0 => 100*sin x = 99*cos x => sin x / cos x = 99/100 => tan x = 99/100 => x = arc tan(99/100) => x = 44.71...

• trigonometry math
We have to prove: (sqrt (1 - (cos t)^2) * sqrt ((sec t )^2) - 1)) / cos t = (tan t)^2 (sqrt (1 - (cos t)^2) * sqrt ((sec t )^2) - 1)) / cos t => (sqrt (sin t)^2 * sqrt ( 1/ (cos t)^2) - 1) / cos...

• trigonometry math
We have to prove that (tan t)^4 + (tan t)^2 + 1 = (1 - (sin t)^2* (cos t)^2) / (cos t)^4 The left hand side: (tan t)^4 + (tan t)^2 + 1 => (sin t)^4 / (cos t)^4 + (sin t)^2 / (cos t)^2 + 1 =>...

• trigonometry math
We have to prove that (csc x)^2 = (2 sec 2x) / (sec 2x - 1) (2 sec 2x) / (sec 2x - 1) => [2*(1/ cos 2x)] / [((1/ cos 2x) - 1)] => [2*(1/ cos 2x)] / [((1 - cos 2x)/ cos 2x)] => 2/ (1 - cos...

• trigonometry math
We are given that sin x = 1/4 and cos x = 3/4. We have to find sec 2x. sec 2x = 1/ cos 2x cos 2x = (cos x)^2 - (sin x)^2 => (3/4)^2 - (1/4)^2 => (9 - 1)/ 16 => 8/ 16 => 1/2 So sec 2x = 2

• trigonometry math
We have to determine the solutions for cos 2a = sqrt( 1 - sin a)^2, lying in the range 0<a<180 cos 2a = sqrt( 1 - (sin a)^2) => cos 2a = sqrt ( (cos a)^2) => cos 2a = cos a => 2*(cos...

• trigonometry math
We have to find the value of [tan (pi/ 7) + cot (pi/2 - pi/6)] / (1+ tan (-pi/7)*tan (pi/6)) We know that cot (pi/2 - pi/6) = tan (pi/6) and as tan is odd tan (-x) = -tan x Also, tan (a + b) =...