### trigonometry math Questions and Answers

trigonometry math

### how to write cos 2x in terms of cos x?

The most straightforward way to obtain the expression for cos(2x) is by using the "cosine of the sum" formula: cos(x + y) = cosx*cosy - sinx*siny. To get cos(2x), write 2x = x + x. Then, cos(2x) =...

trigonometry math

### What is the value of expression cos^2(x-y)+cos^2(x+y)-cos2x*cos2y?

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

### Verify if sin25+sin35=cos5

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

### Evaluate the trigonometric sum sin(pi/3)+sin(2pi/3)+sin(3pi/3)+sin(4pi/3) .

Since this trigonometric sum involves the angles that are multiples of pi/3 (equivalent to 60 degrees)—a "special" angle with known sine, cosine, and tangent values—each of the terms can be...

trigonometry math

### prove tan(a+b)=(tan a+tanb)/(1-tana*tanb)

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

### What are the zeros of the function f(x)=sinx+cosx found in interval (0,pi)?

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

### Calculate sin2a if sina+cosa=1/3

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

### Prove that : sin 3a+sin a+sin 5a/cos 3a+cos a+cos 5a=tan 3a

We are asked to show that (sin3a+sina+sin5a)/(cos3a+cosa+cos5a)=tan3a. Note that sin3a=sina(-1+4cos^2a) and sin5a=sina(1-12cos^2a+16cos^4a)Similarly cos3a=cosa(1-4sin^2a) and...

trigonometry math

### I wish to prove cos^2x-sin^2x=2cos^2x-1.

Another way of solving the problem (cos x)² - (sin x)² = 2(cos x)² - 1 is as follows: Consider the right-hand side (RHS) of the equation: 2(cos x)² - 1 From the Pythagorean Identity, we know that...

trigonometry math

### find the angle x if tan^2 x-4sec x+5=0?

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

### Derive tan(a+b), using sin (a +b) and cos(a +b).

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

### I need to find the solution of equation sin^3x*sin3x+cos^3x*cos3x=0

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

### Calculate tan(x-y), if sin x=1/2 and sin y=1/3. 0<x<90 90<y<180

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

### A triangular plot has a frontage on the sea of 100 yards. The boundary lines running from the beach make angles of 60...

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

### Prove that cos x+sinx /cosx-sinx equal to 1+sin2x/cos2x

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

### trigonometry Prove that cos2x(1 + tanx*tan2x) = 1

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

### Determine the trigonometric form of z=-2i.

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

### Find solutions of the trigonometric equation sin 12x+cos 6x=0?

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

### Solve for x the following 3cos(3x-1)=0.

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

### What is sin(x+y) if sinx+cosy=1/4 and cos x+siny=1/2 ?

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

### What is cos x if tan x=6/11?Solve using right triangle

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

### Find all solutions of the equation in the range [0;pi] . Identify solutions of the equation in [0;pi] . 1 + cosx +...

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

### Simplify the fraction [cosx*sin(2x)-2sinx]/sin^2 x*cosx=?

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

### What is the solution of the equation 2six*sin3x-cos4x=0?

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

### What is the sum of functions sin 135+cos150? please explain the sum

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

### Find the exact solutions of the equation cos 2a=square root(1-sin^2 a), 0<a<180.

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

### show that cos3B+cosB=2(cos2B)cosb.

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

### Solve the equation if x is in interval (0;2pi) cos3x=cosx

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

### Find the number of the solutions of equation 99sin^2x-101=-sin^2x.

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

### Determine the values of x, 0<x<2pi, if the identity is true: cos^6 x=1-sin^6 x

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

### Prove the following identity: (sqrt (1 - (cos^2 ) t ) * sqrt ((sec^2) t - 1)) / cos t = (tan^2) t

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

### Prove AB^2+AC^2=2BM^2+2AM^2 for the diagram at http://i.imgur.com/hOYs3.jpg

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

### Prove that (sin x)^2*[1 + (cot x)^2] = 1

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

### Solve for x in (0 ; 360) : 2 tan x - 1 = -tan (-x).

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

### solve the equation (sin B)^2- 3sin B +2 =0 where 0 ≤ B ≤ 2π

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

### What is t if identity cos2x+2sin^2 (x)+t=0 is true?

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

### equation . Find x if sin^2x=1/4 .

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

### Verify the identity 1+sinx=cos(90-x)+ cot45

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

### Given the right triangle ABC, where A=90 degrees, calculate the expression: E=cosB/sinC+ cosC/sinB

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

### If the identity 1-2sin^2a=2cos^2a+m is true, what is m?

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

### What is x if 100sinx-99cosx=0?

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

### Prove the following identity: (tan^4) t + (tan^2) t + 1 = (1 - (sin^2) t * (cos^2) t) / (cos^4) t

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

### Prove the following identity: (csc^2) x = (2 sec 2x) / (sec 2x - 1)

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

### Given sinx=1/4 and cosx=3/4 what is sec2x ?

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

### What is tan(pi/ 7) + cot(pi/2 - pi/6) / (1+ tan(-pi/7)*tan(pi/6))

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) =...

trigonometry math

### cos problem Given sinx/cosx=2 determine cos^2x.

We have sin x/ cos x = 2 We need to find (cos x)^2 sin x/ cos x = 2 => 2*sin x*cos x = 4*(cos x)^2 => sin 2x = 4*(cos x)^2 => (cos x)^2 = (sin 2x)/4 The value of (cos x)^2 = (sin 2x)/4

trigonometry math

### Verify the inequality : sinx<1/2cosx.

We have to verify sin x < 1 / 2 cos x. sin x < 1 / 2 cos x => 2 sin x * cos x < 1 => sin 2x < 1 But sin 2x can also be equal to 1. Note: Therefore the inequality should be sin x...