**Definition: **Right triangle trigonometry, a central concept in trigonometry, involves the relationship between angles and side ratios in right triangles. Given a right triangle, we can find sine, cosine, and tangent of the non-90º angles. From there, we can also find cosecant, secant, and cotangent.

“SOH-CAH-TOA” is a commonly used mnemonic device for remembering the fundamental trigonometric ratios: **S**ine = **O**pposite over **H**ypotenuse, **C**osine = **A**djacent over **H**ypotenuse, **T**angent = **O**pposite over **A**djacent.

The *hypotenuse* is the side across from the right angle, the *opposite *side refers to the side across from the angle x, and the *adjacent* side refers to the side next to, or touching the angle x.

Keeping this information in mind, we also know:

csc(x) = 1/sin(x) = (hypotenuse/opposite),

sec(x) = 1/cos(x) = (hypotenuse/adjacent), and

cot(x) = 1/tan(x) = (adjacent/opposite).

**Example 1:** Find the six trigonometric ratios for angle x:

**Solution: **

Step 1: Use the Pythagorean Theorem to find the missing side in simplest radical form:

After reducing under the square root, we find the missing side has a length of 6√2.

Step 2: Apply the appropriate ratios, reduce, and rationalize the denominators:

**Example 2: **Find the missing side lengths.

**Solution:**

Step 1: Assess the given information. We know the length of the adjacent side, 5, so to find the length of the hypotenuse, we can use cosine:

cos(x) = adj/hyp

cos(20) = 5/c

Solve for c and use a calculator to evaluate:

c = 5/cos(20) = 5.32, when rounding to the nearest hundredth. This tells us the length of the hypotenuse is 5.32.

Step 2: To find the length of the opposite side, b, we can use tangent:

tan(x) = opp/adj

tan(20) = b/5

Solve for b and use a calculator to evaluate:

b = 5tan(20) = 1.82, when rounding to the nearest hundredth. This tells us the length of side b is 1.82.

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