Use the Pythagorean Theorem to find the length of the missing side.

12^2 + x^2=13^2

144 + x^2 = 169

x^2 = 25 the square root of 25 =5 therefore x =5 which equals the opposite length

Now you have all three lengths

Opposite = 5

Adjacent =12

Hypotenuse = 13

Using SOH CAH TOA

sin(θ) = Opposite/Hypotenuse 5/13

cos(θ) = Adjacent over hypotenuse 12/13.

tan(θ) = opposite over adjacent 5/12

What is SOH CAH TOA?

SOH CAH TOA has been used for years to help students remember the three most common trig ratios? Trig ratios are used in geometry with right triangles to find missing angles or missing sides. The three most common ratios are sin, cosine, and tangent. Let's look at some examples using SOHCAHTOA.

SOH helps one remember that sine equals Opposite over Hypotenuse

CAH helps one remember that cosine equals Adjacent over Hypotenuse

TOA is helpful to remember that tangent equals Opposite over Adjacent

A six foot ladder leans against a building. If the ladder makes a 60∘ angle with the ground. How far up the wall does the ladder reach?

SOH CAH TOA Word Problems

Step 1. Sketch the problem

Step 2. Using SOHCAHTOA determine you can use sin, because you have hypotenuse, and x is opposite.

Step. 3 Sin 60◦ = opposite / hypotenuse

Step 4. Sin 60∘ = x / 6

Step 5. Cross multiply

Step 6. 6 * sin 60∘ = 5.196 feet

SOHCAHTOA Word Problem 2.

A flagpole is 12 feet tall and perpendicular to the ground. A wire extends from the ground to the top of the pole. What is the angle of elevation from the bottom of the wire to the top of the flagpole?

Step 1. Create a drawing

Step 2. Using SOHCAHTOA you can determine you have Sin because you have opposite over hypotenuse

Step 3. In order to find an angle use inverse sin

Step 4. Take the inverse sin of 12 over 20 which equals 36.867

Step 5. Therefore, the angle of elevation is 36.867 degrees