Given that sin(5x-28)degree=cos(3x-5),0 lessthan x lessthan 90degree,find the value of x.
If sin A = cos B, A and B must be complementary angles (if both are less than 90 degrees).
That means
5x - 28 + 3x -5 = 90 degrees
8x = 123
x = 15.375 degrees
Check: sin (5x-28) = sin 48.875 = 0.753276
cos (3x-5) = cos 41.125 = 0.753276
There might be other solutions for which 5x-28 exceeds 90 degrees. Check it out
To solve the equation sin(5x - 28)° = cos(3x - 5) for 0° ≤ x ≤ 90°, we can use trigonometric identities to simplify the equation.
1. Recall that sin(x) = cos(90° - x). We can rewrite the equation as:
sin(5x - 28)° = sin(90° - (3x - 5))
2. Since the sine function is equal for two angles if the two angles themselves are equal or supplementary (add up to 180°), we have:
5x - 28 = 90° - (3x - 5) or 5x - 28 = 180° - (3x - 5)
3. Solve the first equation:
5x - 28 = 90° - 3x + 5
Simplify and rearrange terms:
5x + 3x = 90° + 5 + 28
8x = 123°
x = 123° / 8
x ≈ 15.375°
4. Check the solution:
sin(5x - 28)° = sin(5 * 15.375 - 28)° ≈ sin(68.125)° ≈ 0.934
cos(3x - 5) = cos(3 * 15.375 - 5) ≈ cos(40.125) ≈ 0.934
The values are approximately equal, confirming that x ≈ 15.375° is a valid solution within the specified range.
To find the value of x, we need to solve the equation sin(5x - 28°) = cos(3x - 5°) within the given range of 0° to 90°.
First, let's convert the equation to use the same trigonometric function. We know that sin(x) = cos(90° - x), so we can rewrite the equation as sin(5x - 28°) = sin(90° - (3x - 5°)).
Using the property of equality for sine functions, we set the expressions inside the sin functions equal to each other:
5x - 28° = 90° - (3x - 5°).
Simplifying this equation, we remove the parentheses on the right side:
5x - 28° = 90° - 3x + 5°.
Next, we combine like terms by adding 3x to both sides of the equation:
5x + 3x - 28° = 90° + 5°.
This gives us:
8x - 28° = 95°.
Now we isolate the variable x by adding 28° to both sides:
8x = 123°.
Finally, divide both sides by 8 to solve for x:
x = 123° / 8.
Therefore, the value of x is approximately 15.4°.