Solve the equation
a) Cos (2x +35)0
= sin (3x-10)0
b) Hence, find Tan (2x +4) without using table or calculators.
a) To solve the equation, we can use the Trigonometric identities.
First, let's rearrange the equation:
cos(2x + 35) = sin(3x - 10)
Now, using the identity cos(A) = sin(90 - A), we get:
sin(90 - (2x + 35)) = sin(3x - 10)
Next, we know that sine function is an odd function, which means sin(-x) = -sin(x). Therefore, we can rewrite the equation as:
-sin(2x + 35) = sin(3x - 10)
Now, using the identity sin(A) = -sin(-A), we get:
sin(-(2x + 35)) = sin(3x - 10)
Since sine function is also a periodic function with period of 2π, we can say:
-(2x + 35) = 3x - 10 + 2kπ (where k is an integer)
Now, simplify the equation:
2x + 35 = -3x + 10 - 2kπ
Add 3x to both sides:
5x + 35 = 10 - 2kπ
Subtract 35 from both sides:
5x = -25 - 2kπ
Divide by 5:
x = (-25 - 2kπ)/5
So the general solution to the equation is x = (-25 - 2kπ)/5, where k is an integer.
b) To find tan(2x + 4), we can use the identity:
tan(A) = sin(A) / cos(A)
From the previous equation, we know that cos(2x + 35) = sin(3x - 10). So, substitute this into the equation:
tan(2x + 4) = sin(2x + 35) / cos(2x + 35)
Using the trigonometric identities, we can rewrite:
tan(2x + 4) = -sin(3x - 10) / sin(2x + 35)
Since sin(A) / sin(B) = tan(A - B), we get:
tan(2x + 4) = -tan(3x - 10)
So, the value of tan(2x + 4) is equal to the negative of tan(3x - 10).