Given that sin 2x=cos (3x-10°) , find tan x
since sin(x) = cos(90-x), we have
cos(90-2x) = cos(3x-10)
90-2x = 3x-10
x = 20
tan20° = 0.364
the period is 360/5 = 72°
so x = 20° + 72n°
To find tan x, we need to use the trigonometric identity that relates sine and cosine:
sin^2(x) + cos^2(x) = 1
Let's start by rewriting the given equation using this identity:
sin^2(2x) + cos^2(2x) = 1 [since sin^2(2x) = sin^2(x) and cos^2(2x) = cos^2(x)]
Now, let's work on the right-hand side of the equation:
cos(3x - 10°)
Using the angle sum formula for cosine, we have:
cos(3x)cos(10°) + sin(3x)sin(10°)
Now, let's simplify the equation by using the double-angle formula for sine:
(sin x cos x)cos(10°) + (cos x cos^2 x - sin^2 x sin x)sin(10°)
Expanding further:
sin x cos x cos(10°) + (cos x - sin^2 x)sin(10°)
Now, let's rewrite sin x cos x as 1/2 sin(2x):
(1/2 sin(2x)) cos(10°) + (cos x - sin^2 x)sin(10°)
Next, let's rewrite sin^2 x as 1 - cos^2 x:
(1/2 sin(2x)) cos(10°) + (cos x - (1 - cos^2 x))sin(10°)
Simplifying:
(1/2 sin(2x)) cos(10°) + (cos x - 1 + cos^2 x)sin(10°)
Now, let's substitute this expression back into the original equation:
sin^2(2x) + cos^2(2x) = (1/2 sin(2x)) cos(10°) + (cos x - 1 + cos^2 x)sin(10°)
This equation involves both sin 2x and cos 2x. We need to simplify it further to solve for tan x. However, at this point, it seems difficult to find a direct expression for tan x.
To find the value of tan x, we need to apply trigonometric identities and algebraic manipulations to solve the equation sin 2x = cos (3x - 10°).
1. Start with the given equation: sin 2x = cos (3x - 10°)
2. Apply the double-angle identity for sine: sin 2x = 2sin x cos x
3. Substitute the right-hand side of the equation with the double-angle identity: 2sin x cos x = cos (3x - 10°)
4. Rearrange the equation to have all terms on one side: 2sin x cos x - cos (3x - 10°) = 0
5. Apply the identity for the cosine of the difference of angles: cos (A - B) = cos A cos B + sin A sin B
Now, the equation becomes: 2sin x cos x - cos 3x cos 10° + sin 3x sin 10° = 0
6. Simplify further: sin x (2cos x - cos 3x cos 10°) + cos x (2sin x - sin 3x sin 10°) = 0
7. Factor out sin x and cos x from each term: sin x (2cos x - cos 3x cos 10°) + cos x (2sin x - sin 3x sin 10°) = 0
8. Now, we have two cases to consider:
Case 1: sin x = 0
If sin x = 0, then x = nπ (where n is an integer)
Case 2: sin x ≠ 0
In this case, we can divide both sides of the equation by sin x:
2cos x - cos 3x cos 10° + 2sin x - sin 3x sin 10° = 0
Rearranging the terms: 2(cos x + sin x) = cos 3x cos 10° + sin 3x sin 10°
Simplifying further using the sum-to-product identity: 2(cos x + sin x) = cos (3x - 10°)
Equating the coefficients of cos x and sin x on both sides:
cos x + sin x = 0 ... (1)
cos (3x - 10°) = 2 ... (2)
From equation (1), we have cos x = -sin x
Squaring both sides, we get: cos^2 x = sin^2 x
Applying the identity: sin^2 x + cos^2 x = 1, we can substitute the values to get: 2sin^2 x = 1
Solving for sin x, we get: sin x = ±√(1/2) = ±(1/√2) = ±(√2/2)
Since sin x ≠ 0 (from Case 2), we have:
sin x = (√2/2) or sin x = -(√2/2)
For sin x = (√2/2):
From equation (1), cos x = -sin x = -(√2/2)
Using the relationship: cos^2 x + sin^2 x = 1, we get: cos^2 x + (√2/2)^2 = 1
Simplifying further, we obtain: cos^2 x + 1/2 = 1
Rearranging, we get: cos^2 x = 1/2
Taking the square root, we have: cos x = ±√(1/2) = ±(1/√2) = ±(√2/2)
For cos x = (√2/2):
tan x = sin x / cos x = (√2/2) / (√2/2) = 1
For cos x = - (√2/2):
tan x = sin x / cos x = (√2/2) / (-√2/2) = -1
Therefore, the solutions for tan x are: tan x = ±1.
In summary, the values of tan x are ±1, depending on the specific values of x that satisfy the given equation.