PC LEC 4.3 Right Triangle Trigonometry PDF

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PC LEC 4.3 Right Triangle Trigonometry lecture notes ryan bean MAT 172 spring 2021 precalculus trigonometry SOLUTIONS ALSO POSTED...

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Right Triangle Trigonometry

Section 4.3

Define Trigonometric Functions of Acute Angles In a right triangle with an acute angle , the longest side in the triangle is the hypotenuse (hyp) and is opposite the right angle. The leg that lies on one ray of angle  is called the adjacent leg (adj). The leg that lies across the triangle from  is called the opposite leg (opp).

Hypotenuse (hyp)

Opposite (opp)

Adjacent (adj)

Definition of Trigonometric Functions of Acute Angles Function Name sine cosine tangent cosecant

Definition

Example

opp hyp adj cos   hyp opp tan   adj hyp csc  opp

sin  =

hyp adj adj

sec  =

sin  

secant

sec 

cotangent

cot  

opp

cos  =

25 cm

7 cm

tan  = csc  =

cot  =

24 cm

SOH–CAH–TOA Sine: Opp over Hyp Cosine: Adj over Hyp Tangent: Opp over Adj

Note: the output value of a trigonometric function is unitless because the common units of length “cancel” within each ratio.

Evaluate Trigonometric Functions of Acute Angles 1. First use the Pythagorean theorem to find the length of the missing side. Then find the exact values of sin  and cos .

3

 4

2.

5 tan   12 , then find sec . Given an acute angle , if

Isosceles Right Triangle 45 1

30–60–90 Right Triangle 30

2

2

3

45 1

60

1

 30 = 6

 45 = 4

 60 = 3

sin 

1–2–3

cos 

3–2–1

Trigonometric Function Values of Special Angles



 30 = 6  45 = 4  60 = 3

sin  1 2

cos

tan

csc 

sec 

cot

3 2

3 3

2

2 3 3

3

2 2

2 2 1 2

3 2

1

2 2 3 3

3

2

1

2

3 3

3. Find the exact value without the use of a calculator.

tan a.

   sec 3 4

b.

csc30   2sin 45

Use Fundamental Trigonometric

Reciprocal and Quotient Identities

1 1 sin   sin  or csc  1 1 sec  cos   cos or sec 

sin  and csc are reciprocals.

Pythagorean Identities

cos  and sec  are reciprocals.

sin 2   cos 2  1 tan 2   1 sec2  1  cot 2  csc 2 

1 1 tan   tan  or cot  sin  tan   cos cos cot   sin 

tan  and cot are reciprocals.

csc 

cot  

Identities

tan  is the ratio of sin  and cos  . cot  is the ratio of cos  and sin  .

Cofunction Identities Cofunctions of complementary angles are equal.

sin  cos(90  )   sin  cos     2  tan  cot(90   )   tan  cot     2  sec  csc(90   )   sec  csc     2 

cos 75  4.

Given

cos  sin(90   )   cos  sin     2  cot  tan(90   )   cot  tan     2  csc  sec(90   )   csc  sec     2 

Sine and cosine are cofunctions.

Examples:

Tangent and cotangent are cofunctions.

tan12 cot 78

sin 30 cos 60

csc Secant and cosecant are cofunctions.

5  sec 12 12

6  2 4 , find a cofunction with the same value.

Use the fundamental identities to determine if each expression is true or false. If the statement is false, provide a counterexample. 5.

csc2  tan  sin  cos  1

6.

tan  cos 1  cos

Use Trigonometric Functions in Applications

6.

A 28 foot slide at a water park makes an angle of 60° with the ground as it descends into a pool. What is the vertical distance from the top of the slide to the ground?

7.

An observer on the roof of a 40 ft building measures the angle of depression from the roof to a park bench on the ground to be 24°. What is the distance from the base of the building to the bench as measured along the ground? Round to the nearest foot....