tan(4x − 4)° = cot(3x + 10)°
since tanθ = cot(90-θ),
tan(4x-4) = tan(90-(3x+10))
4x-4 = 80-3x
7x = 84
x = 12
To solve this equation, we need to find the value of x that satisfies the equation tan(4x - 4)° = cot(3x + 10)°.
Step 1: Recall the definitions for tangent and cotangent.
The tangent of an angle is defined as the ratio of the length of the side opposite the angle to the length of the adjacent side.
The cotangent of an angle is defined as the reciprocal of the tangent, which means it is the ratio of the length of the adjacent side to the length of the side opposite the angle.
Step 2: Simplify the equation.
We can start by simplifying the given equation using the reciprocal identities for tangent and cotangent.
Reciprocal identities:
- tan(x) = 1 / cot(x)
- cot(x) = 1 / tan(x)
Using the reciprocal identity for cotangent, we can rewrite the equation as:
tan(4x - 4)° = 1 / tan(3x + 10)°
Step 3: Rewrite tangent in terms of sine and cosine.
To further simplify the equation, we can express tangent in terms of sine and cosine using the identity:
tan(x) = sin(x) / cos(x)
Applying this identity to both sides of the equation, we get:
sin(4x - 4)° / cos(4x - 4)° = 1 / [sin(3x + 10)° / cos(3x + 10)°]
Step 4: Cross-multiply and simplify.
Multiply both sides of the equation by cos(4x - 4)° and sin(3x + 10)° to eliminate the denominators:
sin(4x - 4)° * cos(3x + 10)° = cos(4x - 4)° * sin(3x + 10)°
Step 5: Use the angle sum and difference identities.
Apply the angle sum and difference identities to simplify the equation further.
sin(A + B) = sin(A) * cos(B) + cos(A) * sin(B)
sin(A - B) = sin(A) * cos(B) - cos(A) * sin(B)
Applying the angle difference identity, we can rewrite the equation as:
[sin(4x)° * cos(10)° - cos(4x)° * sin(10)°] = [cos(4x)° * sin(10)° + sin(4x)° * cos(10)°]
Step 6: Simplify the equation.
Combine like terms on both sides of the equation:
sin(4x)° * cos(10)° - cos(4x)° * sin(10)° = cos(4x)° * sin(10)° + sin(4x)° * cos(10)°
The terms cancel out on both sides of the equation:
0 = 2 * sin(4x)° * cos(10)°
Since the property of equality tells us that if a * b = 0, then either a = 0 or b = 0, we have two possibilities:
1) sin(4x)° = 0
2) cos(10)° = 0
Step 7: Find the values of x that satisfy the equation.
1) sin(4x)° = 0
For sin(4x)° = 0 to be true, we know that the angle 4x must equal 0 degrees or a multiple of 180 degrees.
Therefore,
4x = 0
x = 0 / 4
x = 0
2) cos(10)° = 0
To solve this equation, we need to find the values of x that make cos(10)° equal to 0. However, there are no values of x that satisfy this condition, as there is no angle whose cosine is equal to zero.
Thus, the only solution for x in the given equation is x = 0.