Use the cofunction identities to find an angle x that makes the statement true: tan(2x-140) = cot(x+5)
Is the angle in degrees or radians?
degrees
To find an angle x that makes the statement true, we need to use the cofunction identities to rewrite the equation in terms of sine and cosine.
The cofunction identities are:
1. tan(a) = cot(90° - a)
2. cot(a) = tan(90° - a)
Let's use the first cofunction identity to rewrite the equation. We have:
tan(2x - 140°) = cot(x + 5°)
Now, using the cofunction identity, we can rewrite the right side of the equation:
cot(x + 5°) = tan(90° - (x + 5°))
Simplifying further, we have:
cot(x + 5°) = tan(85° - x)
Now we can set the two expressions equal to each other:
tan(2x - 140°) = tan(85° - x)
Since the tangent function is periodic with a period of 180°, we can disregard any differences of multiples of 180° on either side of the equation.
So we have two cases to consider:
Case 1: (2x - 140°) = (85° - x)
This can be solved by adding x to both sides and adding 140° to both sides:
3x - 140° = 85°
3x = 225°
x = 75°
Case 2: (2x - 140°) = (85° - x) + 180°
This can be simplified to:
2x - 140° = 265° - x
Adding x to both sides and adding 140° to both sides:
3x = 405°
x = 135°
Therefore, we have two possible solutions for x: x = 75° and x = 135°.