Pc Right Triangle Trig

Right Triangle Trigonometry

The six trigonometric functions of a right triangle, with an acute angle , are defined by ratios of two sides of the triangle. The sides of the right triangle are:

hyp

the side opposite the acute angle , the side adjacent to the acute angle ,

θ

and the hypotenuse of the right triangle.

adj

The trigonometric functions are sine, cosine, tangent, cotangent, secant, and cosecant. opp sin = cos = adj tan = opp hyp hyp adj csc

=

hyp opp

sec

=

hyp adj

cot

= adj opp

opp

Calculate the trigonometric functions for ∠θ . 5

4

θ 3

The six trig ratios are sin tan sec

4 = 5 4 = 3 5 = 3

cos cot csc

3 = 5 3 = 4 5 = 4

Geometry of the 45-45-90 triangle Consider an isosceles right triangle with two sides of length 1. 45

2

1

12 + 12 = 2

45

1 The Pythagorean Theorem implies that the hypotenuse is of length 2 .

Calculate the trigonometric functions for a 45 angle. 2

1

45

1 opp 1 2 sin 45 = = = hyp 2 2

1 2 adj cos 45 = = = 2 2 hyp

opp 1 tan 45 = = = 1 1 adj

adj 1 cot 45 = = = 1 opp 1

2 hyp sec 45 = = = 2 1 adj

csc 45 =

2 hyp = = 2 opp 1

Geometry of the 30-60-90 triangle Consider an equilateral triangle with each side of length 2.

30○ 30○

The three sides are equal, so the angles are equal; each is 60 .

2

The perpendicular bisector of the base bisects the opposite angle.

60○

Use the Pythagorean Theorem to find the length of the altitude, 3 .

2

3

1

60○

2

1

Calculate the trigonometric functions for a 30 angle. 2

1

30

3 1 opp sin 30 = = hyp 2

3 adj cos 30 = = 2 hyp

3 1 opp tan 30 = = = 3 3 adj

3 adj cot 30 = = = 3 1 opp

2 2 3 hyp sec 30 = = = 3 3 adj

hyp 2 csc 30 = = = 2 opp 1

Calculate the trigonometric functions for a 60 angle. 2

3

60○

opp 3 sin 60 = = hyp 2

1

1 adj cos 60 = = 2 hyp

3 opp tan 60 = = = 3 1 adj

3 1 cot 60 = adj = = 3 3 opp

hyp 2 sec 60 = = = 2 adj 1

2 2 3 hyp csc 60 = = = opp 3 3

Trigonometric Identities are trigonometric equations that hold for all values of the variables. Example: sin θ = cos(90 − θ ), for 0 < θ < 90 Note that θ and 90 − θ are complementary angles. Side a is opposite θ and also adjacent to 90○– θ . a a sin θ = and cos (90 − θ ) = . b b

So, sin θ = cos (90 − θ ).

hyp θ b

90○– θ a

Fundamental Trigonometric Identities for 0 < θ < 90 . Cofunction Identities sin θ = cos(90 − θ ) tan θ = cot(90 − θ ) sec θ = csc(90 − θ )

cos θ = sin(90 − θ ) cot θ = tan(90 − θ ) csc θ = sec(90 − θ )

Reciprocal Identities sin θ = 1/csc θ cot θ = 1/tan θ

cos θ = 1/sec θ sec θ = 1/cos θ

tan θ = 1/cot θ csc θ = 1/sin θ

Quotient Identities tan θ = sin θ /cos θ

cot θ = cos θ /sin θ

Pythagorean Identities sin2 θ + cos2 θ = 1

tan2 θ + 1 = sec2 θ

cot2 θ + 1 = csc2 θ

Example: Given sin θ = 0.25, find cos θ, tan θ, and sec θ. Draw a right triangle with acute angle θ, hypotenuse of length one, and opposite side of length 0.25. Use the Pythagorean Theorem to solve for the third side. 0.25 cos θ = = 0.9682 0.9682 0.9682 tan θ = = 0.258 1 1 sec θ = = 1.033 0.9682

1 θ

0.9682

0.25

Example: Given sec θ = 4, find the values of the other five trigonometric functions of θ . Draw a right triangle with an angle θ such 4

4 hyp that 4 = sec θ = = . adj 1

Use the Pythagorean Theorem to solve for the third side of the triangle. sin θ =

15 4

1 4 15 tan θ = = 15 1

cos θ =

15

θ

1

4 1 = sin θ 15 1 sec θ = =4 cosθ 1 cot θ = 15

csc θ =

Example: Given sin θ = 0.25, find cos θ, tan θ, and sec θ. Draw a right triangle with acute angle θ, hypotenuse of length one, and opposite side of length 0.25. Use the Pythagorean Theorem to solve for the third side. 0.25 cos θ = = 0.9682 0.9682 0.9682 tan θ = = 0.258 1 1 sec θ = = 1.033 0.9682

1 θ

0.9682

0.25