Page 812 - Algebra 1

In addition to the three trigonometric ratios previously discussed, there are three other trigonometric ratios called cosecant, secant, and cotangent, abbreviated csc, sec, and cot, respectively.
Math Reasoning
Generalize Explain the relationship between sine and cosecant; cosine and secant; and tangent and cotangent ratios.
Cosecant, Secant, and Cotangent
___
length of leg opposite ∠A a
cosecant of ∠A = length of hypotenuse = c
_
___
length of leg adjacent to ∠A b
secant of ∠A = length of hypotenuse = c
_
___
length of leg opposite ∠A a
cotangent of ∠A = length of leg adjacent to ∠A = b
_
Example
1
Finding Trigonometric Ratios
Caution
Writing “sin = _5 ” is not 13
a valid trigonometric ratio because there is no angle measure included with sine.
a. Using the right triangle, find sin B, cos B, and tan B. SOLUTION
sin B = opposite leg = 5 ___
hypotenuse 13
cos B = adjacent leg = __ _12
hypotenuse 13
tan B = opposite leg = 5 ___
adjacent leg 12
A
13 5
C 12 B
b. Using the right triangle, find all six trigonometric ratios for ∠A. SOLUTION
First find the length of side a using the Pythagorean Theorem.
B
5
a A 4 C
Caution
Although a = ±3, only the positive value is used because a represents length.
a2 +b2 =c2 a2 +42 =52 a2 +16=25
a2 = 9 a=3
sinA=_a=_3 c5
tanA=_a=_3 b4
secA=_c=_5 b4
cos A = _b = _4 c5
csc A = _c = _5 a3
cot A = _b = _4 a3
Lesson 117 797