Many students begin physics without having been exposed to trigonometry, or having forgotten what they once knew about it. This, however, is not cause for consternation, as the amount of trig needed for this course is easily learned. Trig is based on two Important Facts:

Important Fact #1: The Pythagorean Theorem is True!

1. Use the Pythagorean Theorem to find the missing side on each right triangle:

a)	b)
c)	d)

a) a2 + b2 = c2 (6 cm)2 + (10 cm)2 = c2 c = 11.7 cm

b) a2 + b2 = c2 a2 = c2 - b2 a2 = (12 m)2 - (10 m)2 a = 6.6 m

c) a2 + b2 = c2 (9.4 ft)2 + (17.1 ft)2 = c2 c = 19.5 ft

d) a2 + b2 = c2 a2 = c2 - b2 a2 = (286 in)2 - (97 in)2 a = 269 in

Important Fact #2: Note that as a triangle grows in size, its angles remain the same and its sides retain their proportionality. That is, the ratio of any two sides will be unchanged.

Now you need to learn some terminology. Study the picture below.

See how the angle under consideration is formed by the hypotenuse and an adjacent side. The opposite side gets its name from being opposite the angle. Try your new-found knowledge in the following problems by identifying the hypotenuse, the adjacent side, and the opposite side.

Trig terms are defined in terms of the acute angle you choose.

a)	b)	c)
a) x = Hypotenuse y = Opposite side z = Adjacent side	b) y = Hypotenuse x = Opposite side z = Adjacent side	c) z = Hypotenuse y = Opposite side x = Adjacent side

If you had any trouble with finding the hypotenuse, just remember that it is always the longest side, and is always opposite the right angle.

3. Just to be sure, try one more time on the names of the sides.

a)	b)	c)
a) z = Hypotenuse x = Opposite side y = Adjacent side	b) y = Hypotenuse z = Opposite side x = Adjacent side	c) x = Hypotenuse z = Opposite side y = Adjacent side

Now another new word: The tangent of an angle is the opposite side over the adjacent side. (Keep in mind, we are talking about right triangles, only.)

4. Find the tangents of the angles shown:

a)	b)	c)
a) tan(Ø) = opp/adj tan(Ø) = 4.2 cm/5.6 cm tan(Ø) = .75	b) tan(Ø) = opp/adj tan(Ø) = 19.4 cm/16.8 cm tan(Ø) = 1.15	c) tan(Ø) = opp/adj tan(Ø) = 1.12 cm/.811 in tan(Ø) = (1.12 cm)/[(.811 in)(2.54 cm/in)] tan(Ø) = .54

What about units? Note that since the tangent is the ratio of two lengths, the length measurements will cancel out, leaving the tangent dimensionless.

5. Use the tangent to find the unknown angles:

a)	b)	c)
a) tan(Ø) = (6.7 cm)/(8.6 cm) tan(Ø) = .78 Ø = 37.9°	b) tan(Ø) = (13.5 cm)/[(3.39 in)(2.54 cm/in)] tan(Ø) = 1.57 Ø = 57.5°	c) tan(Ø) = [(9.4 ft)(12 in/1 ft)]/(94.8 in) tan(Ø) = 1.19 Ø = 50.0°

6. Use the tangent to find the missing sides:

a)	b)	c)
a) tan 29° = x/37.2 m x = (37.2 m)(tan 29) x = 20.6 m	b) tan 61° = 11.4 in/x x = (11.4 in)/(tan 61) x = 6.3 in	c) tan 55° = x/96 ft x = (96 ft)(tan 55) x = 137 ft

7*. Find the missing angles:

a)	b)
a) This is a two-step problem. First, let the vertical side be x: tan 34.2 = x/29.1 m x = (29.1 m)(tan 34.2) Now: tan(Ø) = x/31.6 m tan(Ø) = [(29.1 m)(tan 34.2)]/(31.6 m) tan(Ø) = .6258 Ø = 32.0°	b) Again, let the vertical be x: tan 62 = x/12.3 cm x = (12.3 cm)(tan 62) x = 23.13 cm But x is the hypotenuse of the second triangle, now: a2 = x2 - 8.92 a2 = 23.132 - 8.92 a = 21.35 cm tan(Ø) = 8.9/a tan(Ø) = 8.9/21.35 Ø = 22.6°