Right Triangle Trig

Question

I need to find the measure of each angle indicated and round to the nearest tenth. I am given two sides 12 and 13 and one angle C which is 90 degrees. How do I figure this out?

Answer 1

Since you have a triangle with an angle of $90^\circ$, you have a right triangle.

That means you'll want to use the Pythagorean Theorem. (HINT!)

The side opposite the right angle is the longest side: the hypotenuse with length $c$. In your case, we have $c = 13$. The other two sides of the right triangle that meet at the right angle are its legs: say they are of length $a, b$: In your triangle, $a$ is the side opposite angle $A$, and $a$ is unknown; $b$ is the side opposite angle $B$ and $b = 12$. Then we know that $$a^2 + b^2 = c^2\tag{Pythagorean Theorem}$$

Solving for $a$ will give you the length of the side opposite side $A$. To find the measures of the unknown angles, you can use the trigonometric relations given by "TOA SOH CAH" (ask me if you need to, what this means) to determine the two unknown angles.

TOA: $\text{tangent} = \dfrac{\text{opposite}}{\text{adjacent}}$ $\quad \tan A = \dfrac{a}{12}$

SOH: $\text{sine} = \dfrac{\text{opposite}}{\text{hypotenuse}}$ $\quad \sin B = \dfrac{12}{13}$

CAH: $\text{cosine} = \dfrac{\text{adjacent}}{\text{hypotenuse}}\;\;$ E.g.,$\quad \cos B = \dfrac{a}{13}$

You will need only to know one angle, using, say, $B =\sin^{-1}\left(\dfrac{12}{13}\right)$ to find its measure. Then you can solve for $A$ since the sum of the angles of any triangle is standard Euclidean Geometry is $180^\circ$: $$A + B + C = A + B + 90 = 180$$

Answer 2

First use Pythagoreas' theorem to find the side of the last side. By the SOH CAH TOA rule, $\sin \theta=\frac{opposite}{hypotenuse}\implies \sin^{-1}(\theta)=\square $degrees. And then since there are 180 degrees in a triangle, $180-90-\text{(the angle you found)}$= the other angle measure for the last angle. Also if you didn't want to find the last angle that way, you could have just used SOH CAH TOA again, just like you are going to do for the second step of this problem.

Answer 3

You can use laws of sine (http://www.transtutors.com/math-homework-help/laws-of-triangle/)

12/ Sin B = 13 / Sin 90

Sin B = 12 / 13

From here you can calculate Angle B and then using A + B + C = 180 Calculate Angle A