Trig Ratios

1
Finding Trig Ratios

• A trigonometric ratio is a RATIO of the lengths
  of two sides of a RIGHT triangle.
• The three basic trigonometric ratios are called
  sine, cosine, and tangent
• They are abbreviated as sin, cos and tan.

2
Some terminology

• The hypotenuse (hyp) is the longest side of the
  triangle (across from the 90 degree angle) it
  never changes
• The opposite (opp) is the side directly across
  from the angle you are considering
• The adjacent (adj) is the side directly beside
  the angle you are considering that is NOT the
  hypotenuse

3
A picture always helps

• looking at the triangle in terms of angle b

b

• A is the adjacent (near the angle)

C
A

• B is the opposite (across from the angle)

B
b
Near
hyp

• C is always the hypotenuse

Longest
adj
opp
Across
4
But if we switch angles

• looking at the triangle in terms of angle a

• A is the opposite (across from the angle)

C
A
a

• B is the adjacent (near the angle)

B
Across
hyp

• C is always the hypotenuse

Longest
opp
a
adj
Near
5

• Remember we wont use the right angle.
• We will only use the other two acute angles.
• The angle we use is called the reference angle.

X
6
Finding Trig Ratios
SOH-CAH-TOA
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One more thing
8
It is important to note WHICH angle you are
talking about when you find the value of the trig
function.
A
hypotenuse
c
5
b
4
opposite
B
tan B
a
3
sin A
SOH-CAH-TOA
9
Calculating a side if you know the angle
How can we use these?

• you know an angle (25) and its adjacent side
• we want to know the opposite side

10
Ex. 6 Indirect Measurement

• You are measuring the height of a Sitka spruce
  tree in Alaska. You stand 45 feet from the base
  of the tree. You measure the angle of elevation
  from a point on the ground to the top of the top
  of the tree to be 59. To estimate the height of
  the tree, you can write a trigonometric ratio
  that involves the height h and the known length
  of 45 feet.

11
Ex. 7 Estimating Distance

• Escalators. The escalator at the
  Wilshire/Vermont Metro Rail Station in Los
  Angeles rises 76 feet at a 30 angle. To find
  the distance d a person travels on the escalator
  stairs, you can write a trigonometric ratio that
  involves the hypotenuse and the known leg of 76
  feet.

30
12
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13
Why do we need the sin cos?

• We use sin and cos when we need to work with the
  hypotenuse

b
C 10
A
25
B
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And one more sin example
b
C 20
A
25
B
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How do we know which formula to use???

• Well, what are we working with?
• We have an angle
• We have hyp
• We need opp
• With these things we will use the sin formula

C 5
B
65
16
An application

• You look up at an angle of 65 at the top of a
  tree that is 10m away
• the distance to the tree is the adjacent side
• the height of the tree is the opposite side

65
10m
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What are the steps for doing one of these
questions?

1. Make a diagram if needed
2. Determine which angle you are working with
3. Label the sides you are working with
4. Decide which formula fits the sides
5. Substitute the values into the formula
6. Solve the equation for the unknown value
7. Does the answer make sense?