Written in terms of radian measurement, this identity becomes

As mentioned before, we’ll generally use the letter *a* to denote the side opposite angle *A,* the letter *b* to denote the side opposite angle *B,* and the letter *c* to denote the side opposite angle *C.*
Since the sum of the angles in a triangle equals 180°, and angle *C* is 90°, that means angles *A* and *B* add up to 90°, that is, they are complementary angles. Therefore the cosine of *B* equals the sine of *A.* We saw on the last page that sin *A* was the opposite side over the hypotenuse, that is, *a/c.* Hence, cos *B* equals *a/c.* In other words, the cosine of an angle in a right triangle equals the adjacent side divided by the hypotenuse:

where *c* is the hypotenuse. This translates very easily into a Pythagorean identity for sines and cosines. Divide both sides by *c*^{2} and you get

But *a*^{2}/*c*^{2} = (sin *A*)^{2}, and *b*^{2}/*c*^{2} = (cos *A*)^{2}. In order to reduce the number of parentheses that have to be written, it is a convention that the notation sin^{2} *A* is an abbreviation for (sin *A*)^{2}, and similarly for powers of the other trig functions. Thus, we have proven that

when *A* is an acute angle. We haven’t yet seen what sines and cosines of other angles should be, but when we do, we’ll have for any angle *θ* one of most important trigonometric identities, the Pythagorean identity for sines and cosines:

Next consider 30° and 60° angles. In a 30°-60°-90° right triangle, the ratios of the sides are 1 : √3 : 2. It follows that sin 30° = cos 60° = 1/2, and sin 60° = cos 30° = √3 / 2.

These findings are recorded in this table.

Angle | Degrees | Radians | cosine | sine |
---|---|---|---|---|

90° | π/2 | 0 | 1 | |

60° | π/3 | 1/2 | √3 / 2 | |

45° | π/4 | √2 / 2 | √2 / 2 | |

30° | π/6 | √3 / 2 | 1/2 | |

0° | 0 | 1 | 0 |

**30.** *b* = 2.25 meters and cos *A* = 0.15. Find *a* and *c.*

**33.** *b* = 12 feet and cos *B* = 1/3. Find *c* and *a.*

**35.** *b* = 6.4, *c* = 7.8. Find *A* and *a.*

**36.** *A* = 23° 15', *c* = 12.15. Find *a* and *b.*

**30.** The cosine of *A* relates *b* to the hypotenuse *c,* so you can first compute *c.* Once you know *b* and *c,* you can find *a* by the Pythagorean theorem.

**33.** You know *b* and cos *B.* Unfortunately, cos *B* is the ratio of the two sides you don’t know, namely, *a/c.* Still, this gives you an equation to work with: 1/3 = *a/c.* Then *c* = 3*a.* The Pythagorean theorem then implies that *a*^{2} + 144 = 9*a*^{2}. You can solve this last equation for *a* and then find *c.*

**35.** *b* and *c* give *A* by cosines and *a* by the Pythagorean theorem.

**36.** *A* and *c* give *a* by sines and *b* by cosines.

**30.** *c* = *b*/cos *A* = 2.25/0.15 =
15 meters; *a* = 14.83 meters.

**33.** 8*a*^{2} = 144, so *a*^{2} = 18. Therefore *a* is 4.24', or 4'3".
*c* = 3*a* which is 12.73', or 12'9".

**35.** cos *A* = *b/c* = 6.4/7.8 = 0.82. Therefore *A* = 34.86° = 34°52', or about 35°.
*a*^{2} = 7.8^{2} – 6.4^{2} = 19.9, so *a* is about 4.5.

**36.** *a* = *c* sin *A* = 12.15 sin 23°15' = 4.796.
*b* = *c* cos *A* = 12.15 cos 23°15' = 11.17.