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Right Triangle Trig

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Right Triangle Trig

Section 4.3

- 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

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

Opp.

Hyp.

Ѳ

Adj.

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

Ѳ

Adj.

Hyp.

Opp.

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

- Sine = Opposite / Hypotenuse
- Cosine = Adjacent / Hypotenuse
- Tangent = Opposite / Adjacent

S

o

h

C

a

h

T

o

a

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

3

Ѳ

4

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

Ѳ

17

15

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

5

Ѳ

8

- You can also use a trig value to construct a right triangle and find the values of the remaining trig functions.
- E.g. Sin

- Sketch a right triangle and find the values of the remaining trig functions using the given information.
- Cos
- Tan
- Csc

- Cos

- Tan

- Csc

Right Triangle Trig

Section 4.3

- 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

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

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

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

2

- 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.

- From your triangles:

2

=

=

=

=

=

=

- From your triangles:

=

=

2

=

=

=

=

- From your triangles:

=

=

=

=

1

=

=

- 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.

- Using a calculator
- On your calculators, you should see the three main trig functions.
- Using these buttons, find the Sine 10˚

- Your calculator can also evaluate trig functions in radians.
- To do this, you must switch the mode from degrees to radians.
- Find the Cos

- Using your calculator, evaluate the following:
- Tan 67˚
- Sin 3.4
- Sec 35˚
- Cot

- 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˝

- 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

- Evaluate the following trig functions:
- Sin 73˚ 56́
- Tan 44˚ 28́ 16˝
- Sec 4˚ 50́ 15˝

Right Triangle Trig

Section 4.3

- 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

- 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

- 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.

Solve for the remaining parts of the triangle.

8

35˚

Solve for the remaining parts of the triangle.

20˚

12

Solve for the remaining parts of the triangle.

5˚

18

- 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.

Solve for the remaining parts of the triangle.

10

7

Solve for the remaining parts of the triangle.

13

9

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

22˚

7

11

4

7

4

22˚

11

- 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

- Reciprocal Identities

=

=

=

=

=

=

- Quotient Identities

=

=

- Pythagorean Identities

- We use the fundamental identities for 2 main purposes:
- To evaluate trig functions when given certain information
- Transformations (proofs)

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

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

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

- 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.

- Use the fundamental identities to transform the left side of the equation into the right side.
Cos Ѳ Sec Ѳ = 1

- Use the fundamental identities to transform the left side of the equation into the right side.
Cot Ѳ Sin Ѳ = Cos Ѳ

- Use the fundamental identities to transform the left side of the equation into the right side.
(1 + Sin Ѳ) (1 – Sin Ѳ) = Ѳ

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

=

- 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

- 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?

- 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?

- 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.

- 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?