# Right Triangle Trigonometry

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: Sine = Opposite over Hypotenuse, Cosine = Adjacent over Hypotenuse, Tangent = Opposite over Adjacent.

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.

$\dpi{120} \bg_white {\,\,}\textup{SOH-CAH-TOA} \\\\sin\, x = \frac{opposite}{hypotenuse} = \frac{a}{c} \\\\cos\, x = \frac{adjacent}{hypotenuse} = \frac{b}{c} \\\\tan\, x = \frac{opposite}{adjancent} = \frac{a}{b}$

Keeping this information in mind, we also know:

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

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

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:$\dpi{120} \bg_white \\a^2 + b^2 = c^2\\ \\3^2 + b^2 = 9^2\\ \\b^2 = 72\\ \\b = \sqrt{72} = 6\sqrt2$

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:$\dpi{120} \bg_white \\sin\,x = \frac{opp}{hyp} =\frac{3}{9} = \frac{1}{3} \\\\\\cos\,x = \frac{adj}{hyp} =\frac{6\sqrt{2}}{3} = 2\sqrt{2} \\\\\\tan\,x = \frac{opp}{adj} =\frac{6\sqrt{2}}{9} = \frac{2\sqrt{2}}{3} \\\\\\csc\,x = \frac{hyp}{opp} =\frac{9}{3} = 3 \\\\\\sec\,x = \frac{hyp}{adj} =\frac{3}{6\sqrt2} = \frac{1}{2\sqrt{2}} = \frac{\sqrt{2}}{4} \\\\\\cot\,x = \frac{adj}{opp} =\frac{9}{6\sqrt2} = \frac{3}{2\sqrt{2}} = \frac{3\sqrt{2}}{4}$

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(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(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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