Right triangle trig
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Right Triangle Trig. Section 4.3. Right Triangle Trig. In the previous section, we worked with: Angles in both radians & degrees Coterminal angles Complementary and supplementary angles Linear and angular speed Now our focus is going to shift to triangles. Right Triangle Trig.

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Right Triangle Trig

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Right triangle trig

Right Triangle Trig

Section 4.3


Right triangle trig1

Right Triangle Trig

  • In the previous section, we worked with:

  • Angles in both radians & degrees

  • Coterminal angles

  • Complementary and supplementary angles

  • Linear and angular speed

  • Now our focus is going to shift to triangles


Right triangle trig2

Right Triangle Trig

  • In this section, we are going to be using only right triangles

Opp.

Hyp.

Ѳ

Adj.


Right triangle trig3

Right Triangle Trig

  • In this section, we are going to be using only right triangles

Ѳ

Adj.

Hyp.

Opp.


Right triangle trig4

Right Triangle Trig

  • Using these three sides of the right triangle, we can form six ratios that define the six trigonometric functions.


Right triangle trig5

Right Triangle Trig

  • Sine = Opposite / Hypotenuse

  • Cosine = Adjacent / Hypotenuse

  • Tangent = Opposite / Adjacent

S

o

h

C

a

h

T

o

a


Right triangle trig6

Right Triangle Trig

  • Find the value of the six trig functions of the following triangle.

3

Ѳ

4


Right triangle trig7

Right Triangle Trig

  • Find the value of the six trig functions of the following triangle.

Ѳ

17

15


Right triangle trig8

Right Triangle Trig

  • Find the value of the six trig functions of the following triangle.

5

Ѳ

8


Right triangle trig9

Right Triangle Trig

  • You can also use a trig value to construct a right triangle and find the values of the remaining trig functions.

  • E.g. Sin


Right triangle trig10

Right Triangle Trig

  • Sketch a right triangle and find the values of the remaining trig functions using the given information.

  • Cos

  • Tan

  • Csc


Right triangle trig11

Right Triangle Trig

  • Cos


Right triangle trig12

Right Triangle Trig

  • Tan


Right triangle trig13

Right Triangle Trig

  • Csc


Right triangle trig14

Right Triangle Trig

Section 4.3


Right triangle trig15

Right Triangle Trig

  • Yesterday:

    • Defined all six trig functions

    • Found values of all six trig functions from right triangles

    • Constructed right triangles from a specific trig value and found the remaining values

  • Today

    • Special right triangles

    • Using a calculator


Right triangle trig16

Right Triangle Trig

  • Sketch a right triangle and find the values of the remaining trig functions using the given information.

    Cot


Right triangle trig17

Right Triangle Trig

  • Find the sine, cosine, and tangent of 45˚


Right triangle trig18

Right Triangle Trig

  • Using the equilateral triangle below, find the sine, cosine, and tangent of both 30˚ and 60˚.

2


Right triangle trig19

Right Triangle Trig

  • These triangles are our two special triangles.

  • We will use them throughout the year.

  • The sooner you remember them, the easier your life will be.


Right triangle trig20

Right Triangle Trig

  • From your triangles:

2

=

=

=

=

=

=


Right triangle trig21

Right Triangle Trig

  • From your triangles:

=

=

2

=

=

=

=


Right triangle trig22

Right Triangle Trig

  • From your triangles:

=

=

=

=

1

=

=


Right triangle trig23

Right Triangle Trig

  • The trig values of the angles 30˚, 45˚, and 60˚ are values that we will be using from now until May.

  • You will be expected to know these values from memory.

  • There will be quizzes that are non-calculator where these values will be needed.


Right triangle trig24

Right Triangle Trig

  • Using a calculator

  • On your calculators, you should see the three main trig functions.

  • Using these buttons, find the Sine 10˚


Right triangle trig25

Right Triangle Trig

  • Your calculator can also evaluate trig functions in radians.

  • To do this, you must switch the mode from degrees to radians.

  • Find the Cos


Right triangle trig26

Right Triangle Trig

  • Using your calculator, evaluate the following:

  • Tan 67˚

  • Sin 3.4

  • Sec 35˚

  • Cot


Right triangle trig27

Right Triangle Trig

  • In addition to radians and degrees, there is one more type of unit we will be using throughout the year.

  • Minutes and Seconds

    • Most commonly used in longitude and latitude

    • An angle in minutes and seconds would look like:

      • 56˚ 8́ 10˝


Right triangle trig28

Right Triangle Trig

  • To convert minutes and seconds to degrees:

    56˚ 8́ 10˝

    Find the sum of the whole angle, the minutes divided by 60, and the seconds divided by 3,600

    This will give you your angle in degrees


Right triangle trig29

Right Triangle Trig

  • Evaluate the following trig functions:

  • Sin 73˚ 56́

  • Tan 44˚ 28́ 16˝

  • Sec 4˚ 50́ 15˝


Right triangle trig30

Right Triangle Trig

Section 4.3


Right triangle trig31

Right Triangle Trig

  • Find the remaining five trig functions if Tan Ѳ =

  • Find the exact value of the Cos 60˚ and Csc

  • Evaluate the Sec 37˚ to three decimal places


Right triangle trig32

Right Triangle Trig

  • So far:

    • Defined the six trig functions

    • Created triangles from given trig values to find the remaining trig values

    • Used the special right triangles

      • 30-60-9045-45-90

    • Used a calculator to evaluate trig functions of other angles

    • Converted between degrees and minutes/seconds

  • Today

    • Find angles when given trig values

    • Fundamental Identities


Right triangle trig33

Right Triangle Trig

  • So far, we have been using our trig functions to create ratios.

  • We can also use trig functions to solve for entire triangles when given certain information.

  • Must be given 1 side and one other part of the triangle.


Right triangle trig34

Right Triangle Trig

Solve for the remaining parts of the triangle.

8

35˚


Right triangle trig35

Right Triangle Trig

Solve for the remaining parts of the triangle.

20˚

12


Right triangle trig36

Right Triangle Trig

Solve for the remaining parts of the triangle.

18


Right triangle trig37

Right Triangle Trig

  • So far, the information we have been given has been 1 side and 1 angle.

  • When we are given 2 sides, you must use your calculator to evaluate the angle.

  • This involves the inverse trig buttons on your calc.


Right triangle trig38

Right Triangle Trig

Solve for the remaining parts of the triangle.

10

7


Right triangle trig39

Right Triangle Trig

Solve for the remaining parts of the triangle.

13

9


Right triangle trig40

Right Triangle Trig

  • Use the given information to solve for the remaining parts of each triangle.

22˚

7

11

4


Right triangle trig41

Right Triangle Trig

7

4


Right triangle trig42

Right Triangle Trig

22˚

11


Right triangle trig43

Right Triangle Trig

  • Fundamental Trig Identities

    • These are identities that we will use throughout the year

    • It will be very beneficial to memorize them now as opposed to struggling to remember them later

    • We will go over 11 now, there will be over 30 throughout the course of the year

    • These are on page 283 of your book


Right triangle trig44

Right Triangle Trig

  • Reciprocal Identities

=

=

=

=

=

=


Right triangle trig45

Right Triangle Trig

  • Quotient Identities

=

=


Right triangle trig46

Right Triangle Trig

  • Pythagorean Identities


Right triangle trig47

Right Triangle Trig

  • We use the fundamental identities for 2 main purposes:

  • To evaluate trig functions when given certain information

  • Transformations (proofs)


Right triangle trig48

Right Triangle Trig

  • Let Ѳ be an acute angle such that Sin Ѳ = 0.6. Use the fundamental identities to find the Cos Ѳ and Tan Ѳ.


Right triangle trig49

Right Triangle Trig

  • If Tan Ѳ = 5, find the remaining five trig functions of Ѳ.


Right triangle trig50

Right Triangle Trig

  • If CscѲ = , find the remaining 5 trig functions of Ѳ.


Right triangle trig51

Right Triangle Trig

  • Transformations

    • Similar to proofs from geometry

    • Do not need to list reasons, just show steps

    • Can only work with 1 side of the equation

    • For this section, we will only be working with the left side of the equation.


Right triangle trig52

Right Triangle Trig

  • Use the fundamental identities to transform the left side of the equation into the right side.

    Cos Ѳ Sec Ѳ = 1


Right triangle trig53

Right Triangle Trig

  • Use the fundamental identities to transform the left side of the equation into the right side.

    Cot Ѳ Sin Ѳ = Cos Ѳ


Right triangle trig54

Right Triangle Trig

  • Use the fundamental identities to transform the left side of the equation into the right side.

    (1 + Sin Ѳ) (1 – Sin Ѳ) = Ѳ


Right triangle trig55

Right Triangle Trig

  • Use the fundamental identities to transform the left side of the equation into the right side.

=


Right triangle trig56

Right Triangle Trig

  • When dealing with word problems:

    • Start by drawing a picture (preferably relating to the problem)

    • Label all sides of the triangle

    • Determine what information you are looking for

    • Set up a trigonometric ratio and solve


Right triangle trig57

Right Triangle Trig

  • Suppose you are standing parallel to the Concord River. You turn and walk at a 60 degree angle until you reach the river, which is a total distance of 130 ft. How much farther down the river are you than when you started?


Right triangle trig58

Right Triangle Trig

  • You are playing paintball with a friend when you notice him sitting in a tree. The angle of elevation to your friend is 23°. If the distance between yourself and the tree is 60 ft, how high up in the tree is your friend?


Right triangle trig59

Right Triangle Trig

  • Suppose you are putting in a basketball hoop and need to know where to put the “foul line”. You know that the height of the hoop is 10 ft and the angle of elevation from the “foul line” is 33.7°. Use this information to determine where the foul line should go.


Right triangle trig60

Right Triangle Trig

  • You are a surveyor standing 115 ft from the Washington Monument in Washington D.C. You measure the angle of elevation to the top of the monument to be 78.3°. How tall is the Washington Monument?


Right triangle trig61

Right Triangle Trig


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