please help me with some questions I skipped on a review for our test coming up?
solve 5-7 on the interval 0 greater than or equal to x less than or equal to 2pi.
5. sin x=sqrt3/2
6. cos x=-1/2
7. tan x=0
----------------
14. what is the exact value of tan[arcsin(.43)]?
----------------
15-16 find an algebraic expression equivalent to..
15. cos(tan^-1 x)
16. cot(arcsin x)
----------------
solve then describe all solutions
17. tan x-sqrt3=0
18. 2sin x+sqrt3=0
19. 2cos x+sqrt2=0
20. sinx cosx=0
Thank you soo much for any help!
Sure, I'd be happy to help you solve these questions step-by-step! Let's go through each question one by one.
5. To solve sin x = √3/2 on the interval 0 ≤ x ≤ 2π, you can use the inverse sine function (arcsin) to find the value of x. Taking the inverse sine of both sides gives x = arcsin(√3/2). Since sin(π/3) = √3/2, this means x = π/3 or x = 2π/3.
6. To solve cos x = -1/2 on the same interval, you can use the inverse cosine function (arccos). Taking the inverse cosine of both sides gives x = arccos(-1/2). Since cos(2π/3) = -1/2, this means x = 2π/3 or x = 4π/3.
7. To solve tan x = 0 on the interval 0 ≤ x ≤ 2π, you can use the inverse tangent function (arctan). Taking the inverse tangent of both sides gives x = arctan(0). Note that tan(0) = 0, so x = 0.
14. To find the exact value of tan[arcsin(0.43)], you can use the identity tan(arcsin(x)) = x/√(1 - x^2). Substituting x = 0.43, we get tan[arcsin(0.43)] = 0.43/√(1 - 0.43^2). Evaluating this expression gives the exact value of tan[arcsin(0.43)] ≈ 0.4798.
15. To find an algebraic expression equivalent to cos(tan^(-1)(x)), you can use the identity cos(tan^(-1)(x)) = 1/√(1 + x^2). So, the equivalent expression is 1/√(1 + x^2).
16. To find an algebraic expression equivalent to cot(arcsin(x)), you can use the identity cot(arcsin(x)) = √(1 - x^2)/x. So, the equivalent expression is √(1 - x^2)/x.
17. To solve tan(x) - √3 = 0, you can add √3 to both sides to get tan(x) = √3. Then, taking the inverse tangent of both sides gives x = arctan(√3). Since tan(π/3) = √3, this means x = π/3.
18. To solve 2sin(x) + √3 = 0, you can subtract √3 from both sides and divide by 2 to get sin(x) = -√3/2. Then, taking the inverse sine of both sides gives x = arcsin(-√3/2). Since sin(-π/3) = -√3/2, this means x = -π/3 or x = 2π/3.
19. To solve 2cos(x) + √2 = 0, you can subtract √2 from both sides and divide by 2 to get cos(x) = -√2/2. Then, taking the inverse cosine of both sides gives x = arccos(-√2/2). Since cos(3π/4) = -√2/2, this means x = 3π/4 or x = 5π/4.
20. To solve sin(x)cos(x) = 0, you can set each factor equal to 0. This gives two equations: sin(x) = 0 and cos(x) = 0.
For sin(x) = 0, the solutions on the interval 0 ≤ x ≤ 2π are x = 0, x = π, and x = 2π.
For cos(x) = 0, the solutions on the same interval are x = π/2 and x = 3π/2.
So, the solutions to sin(x)cos(x) = 0 are x = 0, x = π/2, x = π, x = 3π/2, and x = 2π.
I hope this helps! Let me know if you have any further questions.
Sure, I'd be happy to help you with these questions! Let's go through each one step by step.
5. The equation is sin x = sqrt(3)/2. To solve this equation on the interval 0 ≤ x ≤ 2π, we need to find the values of x where sin x equals sqrt(3)/2. The values of x that satisfy this equation are π/3 and 2π/3.
6. The equation is cos x = -1/2. To solve this equation on the interval 0 ≤ x ≤ 2π, we need to find the values of x where cos x equals -1/2. The values of x that satisfy this equation are 2π/3 and 4π/3.
7. The equation is tan x = 0. To solve this equation on the interval 0 ≤ x ≤ 2π, we need to find the values of x where tan x equals 0. The values of x that satisfy this equation are 0 and π.
14. The expression tan[arcsin(0.43)] means finding the tangent of the arcsine of 0.43. To find the value, we can first find the angle whose arcsine is 0.43. Using a calculator, we find that arcsin(0.43) ≈ 0.4488 radians. Then we take the tangent of this angle to get tan(0.4488) ≈ 0.4812.
15. The expression cos(tan⁻¹x) means finding the cosine of the arctangent of x. Since these functions are inverse of each other, the value of cos(tan⁻¹x) is simply x.
16. The expression cot(arcsin x) means finding the cotangent of the arcsine of x. We can first find the angle whose arcsine is x. Using a calculator, we find that arcsin(x) ≈ θ. Then we take the cotangent of this angle to get cot(θ).
17. The equation is tan x - sqrt(3) = 0. To solve this equation, we isolate the tangent term by adding sqrt(3) to both sides, which gives tan x = sqrt(3). On the interval 0 ≤ x ≤ 2π, the values of x that satisfy this equation are π/3 and 4π/3.
18. The equation is 2sin x + sqrt(3) = 0. To solve this equation, we isolate the sine term by subtracting sqrt(3) from both sides, which gives 2sin x = -sqrt(3). Then we divide both sides by 2, resulting in sin x = -sqrt(3)/2. On the interval 0 ≤ x ≤ 2π, the values of x that satisfy this equation are 11π/6 and 7π/6.
19. The equation is 2cos x + sqrt(2) = 0. To solve this equation, we isolate the cosine term by subtracting sqrt(2) from both sides, which gives 2cos x = -sqrt(2). Then we divide both sides by 2, resulting in cos x = -sqrt(2)/2. On the interval 0 ≤ x ≤ 2π, the values of x that satisfy this equation are 3π/4 and 5π/4.
20. The equation is sin x cos x = 0. To solve this equation, we use the zero-product property, which states that if the product of two factors is zero, then at least one of the factors must be zero. Therefore, we set sin x = 0 and cos x = 0 separately to find the solutions.
For sin x = 0, the values of x on the interval 0 ≤ x ≤ 2π that satisfy this equation are 0, π, and 2π.
For cos x = 0, the values of x on the interval 0 ≤ x ≤ 2π that satisfy this equation are π/2 and 3π/2.
I hope this helps! Let me know if there's anything else I can assist you with.