7.2 Right Triangle Trigonometry. A triangle in which one angle is a right angle is called a right...

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7.2 Right Triangle Trigonometry

Transcript of 7.2 Right Triangle Trigonometry. A triangle in which one angle is a right angle is called a right...

Page 1: 7.2 Right Triangle Trigonometry. A triangle in which one angle is a right angle is called a right triangle. The side opposite the right angle is called.

7.2

Right Triangle Trigonometry

Page 2: 7.2 Right Triangle Trigonometry. A triangle in which one angle is a right angle is called a right triangle. The side opposite the right angle is called.

A triangle in which one angle is a right angle is called a right triangle. The side opposite the right angle is called the hypotenuse, and the remaining two sides are called the legs of the triangle.

cb

a

90

Page 3: 7.2 Right Triangle Trigonometry. A triangle in which one angle is a right angle is called a right triangle. The side opposite the right angle is called.

Initial side

Terminal side

Page 4: 7.2 Right Triangle Trigonometry. A triangle in which one angle is a right angle is called a right triangle. The side opposite the right angle is called.

a

bc

Page 5: 7.2 Right Triangle Trigonometry. A triangle in which one angle is a right angle is called a right triangle. The side opposite the right angle is called.

The six ratios of a right triangle are called trigonometric functions of acute angles and are defined as follows:

sine of

cosine of

tangent of

cosecant of

secant of

cotangent of

Function namesin

cos

tan

csc

sec

cot

Abbreviation Valueb c

a c

b a

c b

c a

a b

/

/

/

/

/

/

Page 6: 7.2 Right Triangle Trigonometry. A triangle in which one angle is a right angle is called a right triangle. The side opposite the right angle is called.

sin cos

tan csc

sec cot

OppositeHypotenuse

Adjacent

Hypotenuse

OppositeAdjacent

HypotenuseOpposite

HypotenuseAdjacent

AdjacentOpposite

bc

ac

ba

cb

ca

ab

Page 7: 7.2 Right Triangle Trigonometry. A triangle in which one angle is a right angle is called a right triangle. The side opposite the right angle is called.

Find the value of each of the six trigonometric functions of the angle .

Adjacent

12 13

c = Hypotenuse = 13

b = Opposite = 12

a b c2 2 2 a2 2 212 13

a2 169 144 25 a 5

Page 8: 7.2 Right Triangle Trigonometry. A triangle in which one angle is a right angle is called a right triangle. The side opposite the right angle is called.

a

b

c

Adjacent=5

Opposite=

Hypotenuse=

12

13

sin OppositeHypotenuse

1213

cos AdjacentHypotenuse

513

tan OppositeAdjacent

125

csc HypotenuseOpposite

1312

sec HypotenuseAdacent

135

cot AdjacentOpposite

512

Page 9: 7.2 Right Triangle Trigonometry. A triangle in which one angle is a right angle is called a right triangle. The side opposite the right angle is called.

Reciprocal Identities

cscsin

1sec

cos

1

cottan

1

Quotient Identities

tansincos

cotcossin

Page 10: 7.2 Right Triangle Trigonometry. A triangle in which one angle is a right angle is called a right triangle. The side opposite the right angle is called.

b a c2 2 2

cb

a

90

b

c

a

c

c

c

2

2

2

2

2

2

bc

ac

2 2

1

sin cos2 2 1

Page 11: 7.2 Right Triangle Trigonometry. A triangle in which one angle is a right angle is called a right triangle. The side opposite the right angle is called.

tan sec2 21

1 2 2 cot csc

Page 12: 7.2 Right Triangle Trigonometry. A triangle in which one angle is a right angle is called a right triangle. The side opposite the right angle is called.

Given that cos and is an acute angle,

find the exact value of each of the remaining

five trigonometric function of

14

.

sin cos2 2 1

sin221

41

sin22

114

sin

1

14

2

since sin > 0, acute

sin 1116

154

Page 13: 7.2 Right Triangle Trigonometry. A triangle in which one angle is a right angle is called a right triangle. The side opposite the right angle is called.

cos ; sin 14

154

tansincos

15

414

154

41

15

cscsin

1 115

4

415

4 1515

cottan

1 115

1515

seccos

1 114

4

Page 14: 7.2 Right Triangle Trigonometry. A triangle in which one angle is a right angle is called a right triangle. The side opposite the right angle is called.

Theorem Complementary Angles Theorem

Cofunctions of complementary angles are equal.

Page 15: 7.2 Right Triangle Trigonometry. A triangle in which one angle is a right angle is called a right triangle. The side opposite the right angle is called.

sin cos

cos sin

tan cot

cot tan

sec csc

csc sec

90

90

90

90

90

90

(Degrees)

Page 16: 7.2 Right Triangle Trigonometry. A triangle in which one angle is a right angle is called a right triangle. The side opposite the right angle is called.

sin cos

cos sin

tan cot

cot tan

sec csc

csc sec

2

2

2

2

2

2

(Radians)

Page 17: 7.2 Right Triangle Trigonometry. A triangle in which one angle is a right angle is called a right triangle. The side opposite the right angle is called.

sin56 cos 90 56 cos34

sec5

csc

2 5

csc310


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