1.Given that sin x=2/3 and x is an acute angle find without using calculator or mathematical table

Question

a.tan x
b.cos(90-x)
2.find the values of ∅ between 0° and 180° such that 2cos3∅=3sin30
3.if tan x=5/8find sin (90-x) giving your answer in three significant figures
4.solve for x in the equation sin(4x-10)°-cos(x+60)°=0

Answer 1

1. To find tan x, we can use the identity: tan x = sin x / cos x.
Since we know sin x = 2/3, we need to find cos x. We can use the Pythagorean identity: sin^2 x + cos^2 x = 1. Plugging in sin x = 2/3, we get (2/3)^2 + cos^2 x = 1.
Solving for cos x: 4/9 + cos^2 x = 1
cos^2 x = 5/9
Taking the square root of both sides, we get: cos x = ±√(5/9)
Since x is acute, we know that x lies in the first quadrant where cos x is positive.
Therefore, cos x = √(5/9) = √5/3.

Using tan x = sin x / cos x, we can now calculate tan x:
tan x = (2/3) / (√5/3)
tan x = 2/√5

To find cos(90-x), we can use the identity: cos(90-x) = sin x.
Since sin x = 2/3, cos(90-x) = 2/3.

2. The equation 2cos(3∅) = 3sin30 can be simplified using trigonometric identities.
We know that sin 30 = 1/2.
Substitute this value into the equation: 2cos(3∅) = 3(1/2)
2cos(3∅) = 3/2
Divide both sides by 2: cos(3∅) = 3/4

To find the values of ∅, we need to find the angles whose cosine is equal to 3/4.
Using inverse cosine (also known as arccos): ∅ = arccos(3/4).

To find the values of ∅ between 0° and 180°, we can calculate:
∅ = arccos(3/4) ≈ 41.41° and ∅ = 180° - 41.41° ≈ 138.59°

So, the values of ∅ between 0° and 180° such that 2cos(3∅) = 3sin30 are approximately 41.41° and 138.59°.

3. We are given tan x = 5/8.
Using the identity: tan^2 x + 1 = sec^2 x, we can find sec x.
tan x = 5/8.
tan^2 x = (5/8)^2 = 25/64.
sec^2 x = 1 + tan^2 x = 1 + 25/64 = 89/64.
Taking the square root of both sides, we get: sec x = ±√(89/64).

Since x is acute, we know that x lies in the first quadrant where sec x is positive.
Therefore, sec x = √(89/64) = √89/8.

Using the identity: sec^2 x = 1 + tan^2 x, we can find cos^2 x.
sec^2 x = 1 + tan^2 x.
(√89/8)^2 = 1 + (5/8)^2.
89/64 = 1 + 25/64.
89/64 = 89/64.

Using the identity: sin^2 x + cos^2 x = 1, we can find sin x.
sin^2 x = 1 - cos^2 x.
sin^2 x = 1 - 89/64.
sin x = ±√(1 - 89/64) = ±√(64/64 - 89/64) = ±√(-25/64).

Since x is acute, we know that sin x is positive.
Therefore, sin x = √(25/64) = 5/8.

To find sin (90-x), we can use the identity: sin(90-x) = cos x.
Since cos x = 5/8, sin (90-x) = 5/8.

Therefore, sin (90-x) = 5/8.

4. To solve for x in the equation sin(4x-10)° - cos(x+60)° = 0, we need to simplify the equation first.
Using the trigonometric identity: sin(a-b) = sin a cos b - cos a sin b, we can rewrite the equation as:
sin 4x cos 10 - cos 4x sin 10 - cos x cos 60 + sin x sin 60 = 0.
Using the values of cos 10 = √3/2, sin 10 = 1/2, cos 60 = 1/2, sin 60 = √3/2, we get:
(sin 4x * √3/2) - (cos 4x * 1/2) - (cos x * 1/2) + (sin x * √3/2) = 0.

To simplify further, we can remove the denominators by multiplying both sides of the equation by 2:
2(sin 4x * √3) - (cos 4x) - (cos x) + 2(sin x * √3) = 0.

Now, we can group the terms with x and the terms without x:
(2*sin 4x * √3) - (2*sin x * √3) - (cos 4x) - (cos x) = 0.

We can rearrange the terms:
2*sin 4x * √3 - 2*sin x * √3 - cos 4x - cos x = 0.

Factor out √3:
√3(2*sin 4x - 2*sin x) - (cos 4x + cos x) = 0.

Using the identity: sin(a-b) - sin(a+b) = 2*sin a cos b, we can simplify the equation to:
√3(2*sin (4x - x)) - (cos 4x + cos x) = 0.

√3(2*sin 3x) - (cos 4x + cos x) = 0.

Using the identity: cos(a+b) + cos(a-b) = 2*cos a cos b, we can simplify the equation to:
√3(2*sin 3x) - 2*cos (4x + x) = 0.

√3(2*sin 3x) - 2*cos 5x = 0.

Dividing both sides of the equation by 2:
√3*sin 3x - cos 5x = 0.

Now, we can solve for x by finding the values that make this equation true.