14-3 Right Triangles and Trigonometric Ratios

Quick Review

The six different ratios of the sides of a triangle are know as the trigonometric ratios for a right triangle. Those ratios depend on the size of the acute angles in the right triangle.

If θ theta is an acute angle of a right triangle, x is the length of the adjacent leg (ADJ), y is the length of the opposite leg (OPP), and r is the length of the hypotenuse (HYP), then the trigonometric ratios of θ theta are as follows.

sin θ = y r = O P P H Y P cos θ = x r = A D J H Y P tan θ = y x = O P P A D J csc θ = r y = H Y P O P P sec θ = r x = H Y P A D J cot θ = x y = A D J O P P math m l error table with 2 rows and 1 column , row1 column 1 , math m l error table with 3 rows and 2 columns , row1 column 1 , sine theta , column 2 equals , y over r , equals . fraction o p p , over h y p end fraction , row2 column 1 , cosine theta , column 2 equals , x over r , equals . fraction eh d j , over h y p end fraction , row3 column 1 , tangent theta , column 2 equals , y over x , equals . fraction o p p , over eh d j end fraction , end table end math m l error , , row2 column 1 , math m l error table with 3 rows and 2 columns , row1 column 1 , co-secant theta , column 2 equals , r over y , equals . fraction h y p , over o p p end fraction , row2 column 1 , secant theta , column 2 equals , r over x , equals . fraction h y p , over eh d j end fraction , row3 column 1 , co-tangent theta , column 2 equals , x over y , equals . fraction eh d j , over o p p end fraction , end table end math m l error , , end table end math m l error ,

Example

In Δ ABC , ∠ C cap delta is a right angle, a = eh equals 4 and c = c equals 9. What are cos A, sin A, and tan A in fraction form?

Using the Pythagorean Theorem, b = 65 . b equals square root of 65 , .

The ratios are cos A = b c = 65 9 , sin A = a c = 4 9 , cosine eh equals , b over c , equals , fraction square root of 65 , over 9 end fraction . comma . sine eh equals , eh over c , equals , 4 ninths . comma and tan A = a b = 4 65 = 4 65 65 . tangent eh equals , eh over b , equals , fraction 4 , over square root of 65 end fraction , equals . fraction 4 square root of 65 , over 65 end fraction . .

Exercises

Find the values of the six trigonometric functions for the angle in standard position determined by each point.

26. ( − 2 , 4 ) open negative 2 comma 4 close
27. ( − 2 , − 15 ) open negative 2 comma negative 15 close

In Δ ABC , ∠ B cap delta is a right angle, AB = equals 30, and sec A = 5 3 . eh equals , 5 thirds , . Find each value in fraction and in decimal form.

28. cos A
29. sin A
30. tan C
31. csc C

In Δ FGH , ∠ G cap delta is a right angle. Find the remaining sides and angles. Round your answers to the nearest tenth.

32. f = 3 , h = 9 f equals 3 comma h equals 9
33. f = 12 , g = 20 f equals 12 comma g equals 20
34. g = 55 , h = 40 g equals 55 comma h equals 40
35. f = 5 , h = 4 f equals 5 comma h equals 4

Find each length x. Round to the nearest tenth.

14-4 and 14-5 Law of Sines and Law of Cosines

Quick Review

You can use the Law of Sines and the Law of Cosines to find missing measures of a triangle. For Δ ABC : cap delta

The Law of Sines states that sin A a = sin B b = sin C c . fraction sine eh , over eh end fraction . equals . fraction sine b , over b end fraction . equals . fraction sine c , over c end fraction . .

The Law of Cosines

a 2 = b 2 + c 2 − 2 b c cos A eh squared , equals , b squared , plus , c squared , minus 2 b c cosine eh

b 2 = a 2 + c 2 − 2 a c cos B b squared , equals , eh squared , plus , c squared , minus 2 eh c cosine b

c 2 = a 2 + b 2 − 2 a b cos C c squared , equals , eh squared , plus , b squared , minus 2 eh b cosine c

Example

In Δ ABC , m ∠ B = cap delta 60°, a = equals 12, and c = c equals 8. What is b to the nearest tenth?

b 2 = 12 2 + 8 2 − 2 ( 12 ) ( 8 ) cos 60 ° Law of Cosines b 2 = 112 Simplify . b ≈ 10.6 Use a calculator . table with 3 rows and 3 columns , row1 column 1 , b squared , column 2 equals , 12 squared , plus , 8 squared , minus 2 , open 12 close . open 8 close cosine , 60 degrees , column 3 cap lawofcap cosines , row2 column 1 , b squared , column 2 equals 112 , column 3 cap simplify , . , row3 column 1 , b , column 2 almost equal to , 10.6 , column 3 cap useacalculator . . , end table

Exercises

Find the area of each triangle. Round your answers to the nearest hundredth.

38. 
39. 
40. In Δ LMN , m ∠ L = 67 ° , m ∠ N = 24 ° , cap delta and M N = 16 m n equals 16 in. Find LM to the nearest tenth.
41. In Δ DEF , d = 25  in  . , e = 28 cap delta in., and f = 20 f equals 20 in. Find m ∠ F m angle f to the nearest tenth.
42. In Δ GHI , h = 8 , i = 12 , cap delta and m ∠ G = 96 ° . m angle g equals , 96 degrees . Find m ∠ I m angle i to the nearest tenth.

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