
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
tan a + cot a + tan 3a +cot 3a = 8 cos^2 2a/ sin 6a LHS=tan a+cot a + tan (3a) +cot (3a) =tan a + 1/ (tan a )+ tan (3a) + 1/ (tan (3a) ) =(tan^2(a)+1)/(tan(a))+ (tan^2(3a)+1)/(tan(3a))...

Math
The value of `sin((5*pi)/3) ` has to be determined. `sin((5*pi)/3)` = `sin((6*pi)/3  pi/3)` = `sin(2*pi  pi/3)` = `sin pi/3` = `sqrt 3/2` The value of `sin((5*pi)/3) = sqrt 3/2`

Math
You may write the argument of cosine, such that: `23pi/12 = 24pi/12  pi/12` Taking the cosine yields: `cos(23pi/12) = cos(2pi  pi/12) => cos(23pi/12) = cos (pi/12)` Using the half angle...

trigonometry math
One way to solve this problem is to use these identities: sin 2x = 2 sin x*cos x sin x = 2 tan(x/2)/[1+(tan x/2)^2] cos x = [1(tan x/2)^2]/[1+(tan x/2)^2] We'll replace tan x/2 by t: sin x = 2...

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
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
Q : Prove : cot 2x=cosec 2xtanx A : L:H:S ≡ cot 2x = cos 2x / sin 2x = 1 2sin²x / sin 2x { using cos 2x = 12sin²x) = 1/sin 2x  2sin²x / sin 2x = cosec 2x  2sin²x / 2sinx.cosx { using...

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
Well, you could use substitution technique. Let's see why and how to manage this technique. We notice that instead of the 1st term sin 2x, we can use the equivalent product, namely 2 sin x*cos x....

trigonometry math
We'll write tan 2x as a sum of 2 equal angles. tan 2x = tan (x+x) tan 2x = (tan x + tan x)/[1  (tan x)^2] tan 2x = 2tan x/[1  (tan x)^2] (1) We know, from enunciation, the value of tan x = 2/3....

trigonometry math
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trigonometry math
Since x belongs to the range (pi,3pi/2), then x is located in the 3rd quadrant and the values of tangent function are positive. Since the tan function is a ratio between sine and cosine...

trigonometry math
All we need to do is to substitute x by pi/4 in the expresison of the function: f(pi/4) = [cos (pi/4)]^2  sin 2*(pi/4) f(pi/4) = [cos (pi/4)]^2  sin (pi/2) We know that cos pi/4 = (sqrt2)/2 and...

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 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
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
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
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trigonometry math
We'll apply the identity: sin(arccos x) = sqrt(1  x^2) (1) We'll put x = sqrt3/4 We'll raise to square both sides: x^2 = 3/16 We'll substitute x in the expression (1): sin(arccos sqrt3/4) = sqrt(1...

trigonometry math
Both equations are homogenous equations and they may be solved using tangent function. We'll start with the 1st equation: sinx+cosx=0 We'll divide by cos x: sin x*/cos x + 1 = 0 tan x + 1 = 0 tan x...

trigonometry math
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trigonometry math
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trigonometry math
We'll apply tangent functin to the sum of angles: a + b = pi/4 tan (a+b) = tan (pi/4) We'll apply the tangent identity: tan (a+b) = (tan a + tan b)/(1  tan a*tan b) We'll replace tan(pi/4) by it's...

trigonometry math
Before solving the problem, we notice that the angle is foundin the 2nd quadrant, where the values of the function cosine are negative. Now, we'll start solving the problem. We'll use the double...

trigonometry math
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
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
The equation given is 99sin^2x101=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 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
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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 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'll use the identity of triple angle: (sin x)^3 = (3sin x  sin 3x)/4 (cos x)^3 = (cos 3x + 3cos x)/4 We'll substitute th reltions above into equation: sin 3x*(3sin x  sin 3x)/4 + cos 3x*(cos 3x...

trigonometry math
We need to simplify the expression (cos (xy))^2 + (cos (x+y))^2  cos 2x *cos 2y (cos (xy))^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'll rewrite the term (cos z)^2 = [cos (x+y)]^2 cos (x+y) = cos x*cos y  sin x*sin y [cos (x+y)]^2 = (cos x*cos y  sin x*sin y)^2 We'll expand the square: [cos (x+y)]^2 = (cos x*cos y)^2 + (sin...

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
First, we'll rewrite the term 9tan(x), based on the fact that the tangent function is odd, so tg(x)=tgx => 9tan(x) = 9 tan x The equation will become: 10tan x  1  9 tan x = 0 We'll...

trigonometry math
First, we'll move all terms to one side: 2(sin2x+1)/3(sinx+cosx) = 0 We'll multiply by 3 the equation: 2sin2x + 2 + 3(sinx+cosx) = 0 We'll apply the double angle identity for sin 2x: sin 2x =...

trigonometry math
To find the value of the sum, we'll create matching functions in the given sum. For this purpose, we'll substitute the value 0.5 by the equivalent function of the angle pi/6, namely sin pi/6 =...

trigonometry math
We'll use the double angle identity to rewrite the term sin 6x: sin 6x = sin 2*(3x) = 2 sin 3x*cos 3x We'll rewrite the equation, moving all terms to one side: 2 sin 3x*cos 3x + cos 3x = 0 We'll...

trigonometry math
We know that the cosine of double angle can be written as as the cosine of the sum of 2 like angles: cos(x+x) = cos x*cos x  sin x*sin x cos(x+x) = (cos x)^2  (sin x)^2 (1) We'll write cos x in...

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
For the beginning, we'll substitute the function tan pi/4 by it's value 1. We'll transform the sum into a product. For this reason, we'll have to express the value 1 as being the function sine of...

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
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 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 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 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...