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trigonometry math

trigonometry mathTo 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 mathtan 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))...

MathWe wish to find the value of the expression ` ``sin((5pi)/3)` . The domain of `x` over one whole period of the sine function `sin(x)` is ` `0 to `2pi` . If we look at a graph of a single period...

MathWe wish to find the value of the expression `cos((23pi)/12)`. The domain of `x` over one whole period of the cosine function `cos(x)` is ` `0 to `2pi` . If we look at a graph of a single period of...

trigonometry math4 sin 2x + 3 = 4 cos x + 6 sin xHow do i solve this trigonometry question? I can not seem to find...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 mathThe 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 mathWe 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 mathWe'll manage the left side of the given expression: We know that cot (2x) = cos (2x)/sin (2x) We'll apply the double angle identities: cos (2x) = 1  2(sin x)^2 sin (2x) = 2sin x*cos x cot (2x) =...

trigonometry mathFirst, we'll shift cos x to the right side: sqrt3*sin x = cos x We'll divide by cos x both sides: sqrt3*sin x/cos x = 1 But the fraction sin x/cos x can be replaced by the tangent function tan x....

trigonometry mathWe 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 mathWell, 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 mathWe'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 mathWe'll apply the Pythagorean identity to solve the equation: (sin x)^2 + (cos x)^2 = 1 We'll write (sin x)^6 + (cos x)^6 = ((sin x)^2 + (cos x)^2)^3  3(sin x)^2*(cos x)^2)((sin x)^2 + (cos x)^2)...

trigonometry mathSince 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 mathAll 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 mathWe 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 mathWe 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 mathTo 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 mathsin (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 math2x+pi/2 = arccos[cos(xpi/2)] + 2k*pi 2x+pi/2 = (xpi/2)+ 2k*pi We'll subtract pi/2; 2x = x  pi/2  pi/2 + 2k*pi We'll subtract x: x = 2k*pi  pi x1 = pi(2k1) 2x+pi/2 = (xpi/2)+ 2k*pi 2x+pi/2 =...

trigonometry mathWe'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 mathBoth 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 mathThis problem requires the law of cosine to find out the angle V: KG^2 = VG^2 + KV^2  2VG*KV*cos V (1) Since the value of KV is missing, we could find it out, applying the law of cosine for the...

trigonometry mathWe'll shift 2 to the right side: sec 4x = 2 We'll multiply by 1: sec 4x = 2 We'll apply the trigonometric identity: sec 4x = 1/cos 4x We'll rewrite the equation: 1/cos 4x = 2 We'll put 4y = t...

trigonometry mathcot (x+1) = sqrt3 We'll take the arccot function both sides: x + 1 = arccot sqrt 3 + kpi x + 1 = pi/6 + kpi We'll subtract 1 both sides: x = pi/6  1 + kpi The solution of the equation is x = (pi/6...

trigonometry mathWe'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 mathBefore 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 mathWe 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 mathThe 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 mathThe 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 mathWe 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 mathWe know that sin a = sqrt3/2, so that arcsin (sqrt3/2) = a The sine function is positive in the 1st and the 2nd quadrants. a = 60 degrees = pi/3 radians (1st quadrant) a = 120 degrees = 2pi/3...

trigonometry mathWe 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 mathWe 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 mathWe'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 mathWe 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 mathWe'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 mathWe 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 mathFirst, 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 mathFirst, 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 mathTo 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 mathWe'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 mathWe 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 mathWe 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 mathWe 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 mathFor 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 mathThe 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 mathWe 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 mathWe 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 mathWe 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...