ok I'm trying to solve for an angle knowing adjacent and opposite of a triangle
so I do tangent
by definition
tan theta = a^-1 o
how do I algebraically solve for theta
Do you mean adjacent SIDE? and opposite ANGLE?
If so, use the law of sines to get the angle B that is opposite the adjacent side (b).
Let A be the angle for which you know the adjacent side length, a. Then
a/sin A = b/sin B
solve for sin B and angle B
Then use C = 180 - A - B degrees
There will be two possible values of B if obtuse angles are allowed. If so, you have been given the ambiguous side-side-angle (SSA) set of known quantities.
To solve for an angle using the tangent function, you can follow these steps algebraically:
1. Start with the equation tan(theta) = opposite/adjacent (tan theta = o/a).
2. To isolate the angle theta, take the inverse tangent (arctan) of both sides: theta = arctan(opposite/adjacent).
3. Use a scientific calculator or a table of tangent values to find the inverse tangent (arctan) of the ratio opposite/adjacent. By inputting the ratio and applying the arctan function, you will obtain the angle in radians.
A common mistake to avoid is mixing up the order of the sides when plugging in the values into the arctan function. Ensure that the opposite side is divided by the adjacent side in the correct order.
Note: The result from the inverse tangent function will give you the angle in radians. If you want the angle in degrees, convert it by multiplying the result by (180/pi).