Unit 4, lesson 49

Right Triangle Trigonometry

How the sides of a right triangle define sine, cosine, tangent and their reciprocals, why these ratios depend only on the angle, and how to use them to find heights and distances.

  • Theory part 97
  • Practice part 98
  • Triangles and the Unit Circle

You can find the height of a tree without climbing it. Stand a measured distance from the trunk, measure the angle up to the top, and a right triangle does the rest. This works because every right triangle with that same angle has the same shape, so the ratios of its sides depend only on the angle. Those ratios are the trigonometric ratios.

Key terms

Right triangle

Definition A right triangle is a triangle with one right angle, an angle of 90∘90^\circ or π2\frac{\pi}{2} radians. The angles of a triangle add up to 180∘180^\circ, so the other two angles are acute and add up to 90∘90^\circ. The two sides that form the right angle are the legs. If the legs have lengths aa and bb and the third side has length cc, the Pythagorean theorem says

a2+b2=c2.a^2 + b^2 = c^2.

Conversely, a triangle whose side lengths satisfy a2+b2=c2a^2 + b^2 = c^2 has a right angle opposite the side of length cc.

In plain words One corner of the triangle is square, like the corner of a sheet of paper. Know any two sides and the Pythagorean theorem gives you the third.

Example A right triangle has legs 5 and 12. Its third side is

c=52+122=169=13.c = \sqrt{5^2 + 12^2} = \sqrt{169} = 13.

Only the positive square root counts, because a length is positive.

Hypotenuse

Definition The hypotenuse of a right triangle is the side opposite the right angle. It is the longest side. With legs aa and bb and hypotenuse cc, the Pythagorean theorem gives c2=a2+b2>a2c^2 = a^2 + b^2 > a^2, so c>ac > a, and in the same way c>bc > b.

In plain words The hypotenuse is the slanted side across from the square corner. It is the one side that does not touch the right angle.

Example A ladder stands on level ground and leans against a vertical wall, with its foot 1.51.5 meters from the wall and its top 3.63.6 meters up. The wall, the ground and the ladder form a right triangle with the ladder as the hypotenuse, so the ladder is

1.52+3.62=2.25+12.96=15.21=3.9 meters long.\begin{aligned} \sqrt{1.5^2 + 3.6^2} &= \sqrt{2.25 + 12.96} \\[0.4em] &= \sqrt{15.21} \\[0.4em] &= 3.9 \text{ meters long.} \end{aligned}

Opposite and adjacent sides

Definition Fix one acute angle θ\theta of a right triangle. The side opposite θ\theta is the leg that does not touch the vertex of θ\theta. The side adjacent to θ\theta is the leg that does touch it. The hypotenuse also touches that vertex, but it is never called the adjacent side. These labels depend on the angle you choose, because the leg opposite one acute angle is adjacent to the other.

In plain words Stand at the corner with angle θ\theta. The leg across the triangle from you is opposite, and the leg running along your corner is adjacent.

Example Take triangle ABCABC with the right angle at CC, legs BC=5BC = 5 and AC=12AC = 12, and hypotenuse AB=13AB = 13.

AngleOpposite sideAdjacent sideHypotenuse
AABC=5BC = 5AC=12AC = 12AB=13AB = 13
BBAC=12AC = 12BC=5BC = 5AB=13AB = 13

The two legs swap roles when you move from angle AA to angle BB.

Similar triangles

Definition Two triangles are similar when their angles can be matched so that corresponding angles are equal. This happens exactly when their corresponding sides are proportional, which means there is a number k>0k > 0 such that each side of the second triangle is kk times the matching side of the first. Two pairs of equal angles are enough, since the third angles then agree as well. In particular, two right triangles with one equal acute angle are similar. Ratios of sides inside a triangle do not change under scaling, because

kakc=acfor k>0 and c>0.\frac{ka}{kc} = \frac{a}{c} \qquad \text{for } k > 0 \text{ and } c > 0.

In plain words Similar triangles have the same shape and possibly different sizes. Scaling multiplies every side by the same factor, so the ratio of any two sides stays the same.

Example The right triangles with sides 3, 4, 5 and 6, 8, 10 are similar with k=2k = 2. The shortest side divided by the hypotenuse is 35=610=0.6\frac{3}{5} = \frac{6}{10} = 0.6 in both, and the shortest side divided by the middle side is 34=68=0.75\frac{3}{4} = \frac{6}{8} = 0.75 in both.

Six trigonometric ratios

Definition Let θ\theta be an acute angle of a right triangle, so 0∘<θ<90∘0^\circ < \theta < 90^\circ. Write opp, adj and hyp for the lengths of the side opposite θ\theta, the side adjacent to θ\theta and the hypotenuse. The six trigonometric ratios of θ\theta are

sin⁡θ=opphyp,cos⁡θ=adjhyp,tan⁡θ=oppadj,\begin{aligned} \sin \theta &= \frac{\text{opp}}{\text{hyp}}, \\[0.8em] \cos \theta &= \frac{\text{adj}}{\text{hyp}}, \\[0.8em] \tan \theta &= \frac{\text{opp}}{\text{adj}}, \end{aligned}csc⁡θ=hypopp,sec⁡θ=hypadj,cot⁡θ=adjopp.\begin{aligned} \csc \theta &= \frac{\text{hyp}}{\text{opp}}, \\[0.8em] \sec \theta &= \frac{\text{hyp}}{\text{adj}}, \\[0.8em] \cot \theta &= \frac{\text{adj}}{\text{opp}}. \end{aligned}

They are called sine, cosine, tangent, cosecant, secant and cotangent. By similarity, each ratio depends only on θ\theta and not on the size of the triangle. Cosecant, secant and cotangent are the reciprocals of sine, cosine and tangent. Since the hypotenuse is the longest side, 0<sin⁡θ<10 < \sin \theta < 1 and 0<cos⁡θ<10 < \cos \theta < 1 for every acute angle θ\theta.

In plain words Each ratio is one side of the triangle divided by another. Many students remember the first three with SOH CAH TOA, short for sine is opposite over hypotenuse, cosine is adjacent over hypotenuse and tangent is opposite over adjacent. The other three are those fractions turned upside down.

Example In the triangle with legs 5 and 12 and hypotenuse 13, take θ\theta to be the angle opposite the side 5. Then opp is 5, adj is 12 and hyp is 13.

sin⁡θ\sin \thetacos⁡θ\cos \thetatan⁡θ\tan \thetacsc⁡θ\csc \thetasec⁡θ\sec \thetacot⁡θ\cot \theta
513\frac{5}{13}1213\frac{12}{13}512\frac{5}{12}135\frac{13}{5}1312\frac{13}{12}125\frac{12}{5}

For the other acute angle the numbers 5 and 12 trade places, so its sine is 1213\frac{12}{13}, which equals cos⁡θ\cos \theta.

Angle of elevation

Definition When an observer looks up at an object, the angle of elevation is the angle between the line of sight to the object and the horizontal through the observer’s eye, measured upward in the vertical plane that contains the line of sight. For an object above eye level that is not straight overhead, it lies strictly between 0∘0^\circ and 90∘90^\circ, and that is the case in every right triangle problem.

In plain words Look straight ahead, then tilt your gaze up until you see the object. The amount of tilt is the angle of elevation.

Example You stand on level ground 20 meters from the base of an upright tree. From your eye, 1.61.6 meters above the ground, the angle of elevation of the treetop is 38∘38^\circ. The 20 meters is adjacent to that angle and the height of the treetop above your eye is opposite it, so that height is 20tan⁡38∘20 \tan 38^\circ. The whole tree is

h=1.6+20tan⁡38∘≈1.6+15.63≈17.2 meters tall.\begin{aligned} h &= 1.6 + 20 \tan 38^\circ \\[0.4em] &\approx 1.6 + 15.63 \\[0.4em] &\approx 17.2 \text{ meters tall.} \end{aligned}

A common slip in problems like this is to leave out the eye height.

Angle of depression

Definition When an observer looks down at an object, the angle of depression is the angle between the line of sight down to the object and the horizontal through the observer’s eye, measured downward in the vertical plane that contains the line of sight. For an object below eye level that is not straight below, it lies strictly between 0∘0^\circ and 90∘90^\circ. In that plane the horizontal lines through the observer and through the object are parallel and the line of sight crosses both, so the angle of depression from the observer to the object equals the angle of elevation from the object to the observer, as alternate interior angles.

In plain words The angle of depression is measured down from the horizontal, not from a vertical wall or cliff. Inside the right triangle it reappears at the bottom corner as an angle of elevation.

Example From a lighthouse window 45 meters above sea level, the angle of depression to a boat is 12∘12^\circ. At the boat the angle of elevation of the window is also 12∘12^\circ, with the height 45 meters opposite it and the distance dd to the base of the lighthouse adjacent. So tan⁡12∘=45d\tan 12^\circ = \frac{45}{d}, and

d=45tan⁡12∘≈450.2126≈211.7 meters.\begin{aligned} d &= \frac{45}{\tan 12^\circ} \approx \frac{45}{0.2126} \\[0.8em] &\approx 211.7 \text{ meters.} \end{aligned}

Contextual units

Definition A trigonometric ratio divides one length by another length measured in the same unit, so it is a pure number with no unit. When you solve an equation such as opp=hyp⋅sin⁡θ\text{opp} = \text{hyp} \cdot \sin \theta for a length, the answer carries the unit of the given length. Two habits follow. Convert all lengths to one unit before you divide them, and make sure the angle unit matches the calculator mode, degree mode for an angle in degrees and radian mode for an angle in radians. State a final answer with its unit and a sensible rounding.

In plain words The ratios have no units but the sides do. Give answers in the units of the problem, and check that your calculator reads the angle the way you meant it.

Example A 4 meter ladder makes an angle of 75∘75^\circ with level ground. Its top is at height

4sin⁡75∘≈4⋅0.9659≈3.86 meters.\begin{aligned} 4 \sin 75^\circ &\approx 4 \cdot 0.9659 \\[0.4em] &\approx 3.86 \text{ meters.} \end{aligned}

In radian mode the same keystrokes compute sin⁡75≈−0.3878\sin 75 \approx -0.3878, the sine of 75 radians, and the height comes out as about −1.55-1.55 meters. A negative length is a clear sign that the mode was wrong.

Check yourself

  1. A right triangle has legs 8 and 15. Find the hypotenuse and all six trigonometric ratios of the acute angle θ\theta opposite the leg of length 8.
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The hypotenuse is 64+225=289=17\sqrt{64 + 225} = \sqrt{289} = 17. With opp 8, adj 15 and hyp 17,

sin⁡θ=817,cos⁡θ=1517,tan⁡θ=815,\begin{aligned} \sin \theta &= \frac{8}{17}, \\[0.8em] \cos \theta &= \frac{15}{17}, \\[0.8em] \tan \theta &= \frac{8}{15}, \end{aligned} csc⁡θ=178,sec⁡θ=1715,cot⁡θ=158.\begin{aligned} \csc \theta &= \frac{17}{8}, \\[0.8em] \sec \theta &= \frac{17}{15}, \\[0.8em] \cot \theta &= \frac{15}{8}. \end{aligned}
  1. A kite string 60 meters long is pulled straight. The hand holding it is 1 meter above level ground, and the angle of elevation of the kite from the hand is 50∘50^\circ. How high above the ground is the kite?
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The string is the hypotenuse and the height above the hand is opposite the 50∘50^\circ angle, so that height is 60sin⁡50∘≈45.9660 \sin 50^\circ \approx 45.96 meters. Adding the hand height gives

1+60sin⁡50∘≈47.0 meters.1 + 60 \sin 50^\circ \approx 47.0 \text{ meters.}
  1. A right triangle has hypotenuse 1 and an acute angle θ\theta whose opposite side has length xx, where 0<x<10 < x < 1. Write cos⁡θ\cos \theta and tan⁡θ\tan \theta in terms of xx.
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By the Pythagorean theorem the adjacent side is 1−x2\sqrt{1 - x^2}, which is positive because 0<x<10 < x < 1. So

cos⁡θ=1−x2andtan⁡θ=x1−x2.\begin{aligned} \cos \theta &= \sqrt{1 - x^2} \\[0.8em] \text{and} \quad \tan \theta &= \frac{x}{\sqrt{1 - x^2}}. \end{aligned}

How this lesson connects