Trigonometric Equations And Trigonometric Identities Quiz

Explanation
The equation given is 2sin(-x) = 0.3. Since the sine function is periodic with a period of 360 degrees, we can find the angles between 0 and 360 degrees that satisfy the equation. By taking the inverse sine of both sides and solving for x, we find that x is equal to -sin^(-1)(0.15) and 180 - sin^(-1)(0.15). These values can be approximated to 188.6 and 351.4 degrees respectively, which matches the given answer.

Explanation
The equation given is 3sin(x) + 2 = tan(75). To solve for x, we need to isolate sin(x) on one side of the equation. Subtracting 2 from both sides gives us 3sin(x) = tan(75) - 2. Next, divide both sides by 3 to get sin(x) = (tan(75) - 2)/3. Using a calculator, we find that tan(75) is approximately 3.732. Substituting this value into the equation gives sin(x) = (3.732 - 2)/3, which simplifies to sin(x) = 0.577. To find the angles between 0 and 360 degrees that have a sine value of 0.577, we can use inverse sine function. The angles are approximately 35.3 and 144.7 degrees.

Explanation
The equation 5 cos x + 2 sin x = 0 can be rewritten as sin x / cos x = -5/2. By using the identity tan x = sin x / cos x, we can find the values of x that satisfy the equation. Taking the inverse tangent of -5/2, we get x = -68.2 degrees and x = 111.8 degrees. However, since we are looking for angles between 0 and 360 degrees, we need to add 180 degrees to both solutions. Therefore, x = 111.8 degrees and x = 291.8 degrees satisfy the equation.

Explanation
The equation tan(x-60 degrees) = 1/SQRT3 can be rewritten as tan(x) = 1/SQRT3 when we add 60 degrees to both sides. The value of 1/SQRT3 is approximately 0.577. We know that the tangent function is positive in the first and third quadrants, so we need to find the angles where tan(x) = 0.577. The angles that satisfy this equation are approximately 30 degrees and 150 degrees. However, since we added 60 degrees to both sides, the values of x that satisfy the original equation are 90 degrees and 210 degrees, which can be rounded to one decimal place as 90 and 270.

Explanation
The equation cot (2x + 10 degrees) = -0.5 can be solved by finding the angles whose cotangent is -0.5. The cotangent function is negative in the second and fourth quadrants. By using the inverse cotangent function, we can find the angles in these quadrants that have a cotangent of -0.5. The angles in the second quadrant can be found by subtracting 10 degrees from multiples of 180 degrees. Similarly, the angles in the fourth quadrant can be found by adding 10 degrees to multiples of 180 degrees. The solutions are 53.3, 143.3, 233.3, and 323.3 degrees.

Explanation
The equation given is 2 tan x = 4 - sec^2 x. To solve this equation, we can start by rearranging it to sec^2 x + 2 tan x - 4 = 0. We can then substitute sec^2 x = 1 + tan^2 x into the equation to get tan^2 x + 2 tan x - 3 = 0. Factoring this equation gives us (tan x + 3)(tan x - 1) = 0. Therefore, tan x = -3 or tan x = 1. The angles that satisfy these conditions are x = 45 degrees, 108.4 degrees, 225 degrees, and 288.4 degrees.

Explanation
The equation 2 sin x cos x = tan x can be simplified using trigonometric identities. By dividing both sides of the equation by cos^2 x, we get 2 sin x = tan x / cos x. Using the identity sin x = tan x / sec x, we can substitute and simplify further to 2 sin x = sin x. This equation holds true for any value of x, so all angles between 0 and 360 degrees inclusive satisfy the equation. The given answer x=0,45,135,180,225,315,360 lists all the angles that satisfy the equation.

Explanation
The given equation is 1 + ______ = sec^2x * tan^2x. By simplifying the equation, we can see that the missing term is tan^2x. This can be determined by using the trigonometric identity sec^2x = 1 + tan^2x, which states that the square of the secant function is equal to 1 plus the square of the tangent function. Therefore, the correct answer is tan^2x.

Explanation
The given equation cos x = -0.71 represents the cosine function, which gives the ratio of the adjacent side to the hypotenuse in a right triangle. The value -0.71 is negative, indicating that the angle x is in the second or third quadrant. The angle x = 135.2 is the only angle between 0 and 360 degrees that satisfies this equation and has one decimal place. Therefore, the answer is x = 135.2.