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Acute angles, right triangles, and trigonometry

Written by tutor Jessica G.

An acute angle is one whose measure is less than 90 degrees. An acute triangle, therefore, is a triangle whose three angles each measure less than 90 degrees. An equilateral triangle is a specific type of acute triangle where the three angles have an equal measure of 180°/3 = 60°.

A right angle, formed by two intersecting perpendicular lines, measures 90 degrees. Right triangles contain one 90 degree angle. Because each triangle’s interior angles sum to 180 degrees, a right triangle can only contain one right angle. (Similarly, an obtuse triangle can only contain one obtuse angle.)

Within a right triangle, we have three basic trigonometric ratios that we study: sine (abbreviated sin), cosine (abbreviated cos), and tangent (abbreviated tan). These three trigonometric ratios relate the sides of a triangle to the non-right angles in a right triangle, or, as we call them, acute angles. We refer to our acute angle as theta (θ), and label the sides of triangle in relation to theta as follows. Recall that the angle opposite the right angle of a triangle is called the hypotenuse, and the other two sides of the triangle (those that form the right angle) are the legs of the triangle.

We refer to the legs now as opposite (across the triangle from) theta and adjacent (next to) to theta. These labels are important when we write our trigonometric ratios. Each acute angle has a set of three unique trigonometric ratios, sine, cosine, and tangent. (See law of sines and law of cosines for applications in obtuse and non-right acute triangles.) Now that we have each of our triangle sides uniquely labeled, we can identify our trigonometric ratios truly as ratios of sides of our triangle:

cos(θ) = adjacent/hypotenuse
sin(θ) = opposite/hypotenuse
tan(θ) = opposite/adjacent

There are many mnemonic devices for remembering the trigonometric ratios, but these two are most common:

SohCahToa (pronounced sew-cuh-toe-uh)

Oscar/Had = Some
A/Heap = Corn
Of/Apples = Too

(read down the ratios, as Oscar Had A Heap of Apples, Some Corn Too)

These ratios help us solve two main types of problems: solving for a missing side length when we’re only given one and not able to use they Pythagorean theorem (example 1 below), and solving for a missing angle when we’re only given the right angle and cannot therefore subtract from 180° (example 2 below).

Example 1: Given m∠B = 43° and AB = 7, find the length of AC.

Since we have a side length for the hypotenuse and we’re looking for the length of the opposite leg, we need to use the trigonometric ratio sine to solve. Substituting the values we know and calling the length of AC x, we have sin(43)=x/7. If we multiply both sides by 7, we have 7sin(43)=x/7·7/1 which gives 7sin(43) = x. Plugging the left-hand side of the equation into our calculators, we get x=4.77398852, so the length of AC is approximately 4.8. (If you're using a graphing calculator, make sure it's in degree mode.)

Example 2: Given BC = 7 and AC = 11, find m∠B

Since we have the lengths of the adjacent and opposite legs, in relation to the angle we’re looking for, we need to use the trigonometric ratio tangent. Substituting the values we know, we have tan(θ)=11/7. Now, our calculator will only let us find the tangent of an angle, but since we’re looking for the angle measurement, we need to take the inverse of the tangent function (for a review of inverse functions, see the related help page). We know that when we take an inverse, we “switch” the argument (in this case theta) and the function value (here 11/7). We therefore now have tan-1(11/7) = θ. Since we now have a numerical value inside our tangent function, we can use our calculator to evaluate (on the TI calculators, use the 2nd key, then the TAN key) and get 57.52880771 = θ, so m∠B ≈ 57.5°.

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