Right Angled Trigonometry

1
Right Angled Trigonometry
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Labeling a Right Triangle

• In trigonometry, we give each side a name
  according to its position in relation to any
  given angle in the triangle Hypotenuse,
  Opposite, Adjacent

• The _________ is always the longest side of the
  triangle.

hypotenuse
Hypotenuse
Adjacent

• The _________ side is the leg directly across
  from the angle.

opposite
Opposite

• The _________ side is the leg alongside the angle.

adjacent
3
Trigonometric Ratios

• We define the 3 trigonometric ratios in terms of
  fractions of sides of right angled triangles.

•  

Hypotenuse (HYP)
Adjacent (ADJ)
Opposite (OPP)
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SohCahToa

• Sine equals Opposite over Hypotenuse
• Cosine equals Adjacent over Hypotenuse
• Tangent equals Opposite over Adjacent

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Practice Together

• Given each triangle, write the ratio that could
  be used to find x by connecting the angle and
  sides given.

32?
b
x
6
YOU DO

• Given the triangle, write all the ratios that
  could be used to find x by connecting the angle
  and sides given.

c
7
In a right triangle, if we are given another
angle and a side we can find

• The third angle of the right triangle
• How?
• The other sides of the right triangle
• How?

Using the angle sum of a triangle is 180?
Using the trigonometric ratios
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Steps to finding the missing sides of a right
triangle using trigonometric ratios

1. Redraw the figure and mark on it HYP, OPP, ADJ
   relative to the given angle

HYP
ADJ
OPP
9
Steps to finding the missing sides of a right
triangle using trigonometric ratios

• For the given angle choose the correct
  trigonometric ratio which can be used to set up
  an equation
• Set up the equation

HYP
ADJ
 
OPP
 
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Steps to finding the missing sides of a right
triangle using trigonometric ratios

• Solve the equation to find the unknown.

HYP
ADJ
 
 
OPP
 
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Practice Together

• Find, to 2 decimal places, the unknown length in
  the triangle.

12
YOU DO

• Find, to 1 decimal place, all the unknown angles
  and sides in the triangle.

b m
63?
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Steps to finding the missing angle of a right
triangle using trigonometric ratios

1. Redraw the figure and mark on it HYP, OPP, ADJ
   relative to the unknown angle

OPP
ADJ
2.67 km
HYP
14
Steps to finding the missing angle of a right
triangle using trigonometric ratios

• For the unknown angle choose the correct trig
  ratio which can be used to set up an equation
• Set up the equation

OPP
ADJ
2.67 km
 
HYP
 
15
Steps to finding the missing angle of a right
triangle using trigonometric ratios

1. Solve the equation to find the unknown using the
   inverse of trigonometric ratio.

OPP
 
ADJ
2.67 km
HYP
 
 
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Practice Together

• Find, to one decimal place, the unknown angle in
  the triangle.

17
YOU DO

• Find, to 1 decimal place, the unknown angle in
  the given triangle.

4 m
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Practice Isosceles Triangles

• Using what we already know about right angles in
  isosceles triangles find the unknown side.

x cm
67?
10 cm
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YOU DO Isosceles Triangles

• Find the unknown angle of the isosceles triangle
  using what you already know about right angles in
  isosceles triangles.

5.2 m
8.3 m
??
20
Practice Circle Problems

• Use what you already know about right angles in
  circle problems to find the unknown angle.

10 cm
??
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YOU DO Circle Problems

• Use what you already know about right angles in
  circle problems to find the unknown side length.

6.5 cm
56?
x cm
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Practice Other Figures (Trapezoid)

• Find x given

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YOU DO Other Figures (Rhombus)

• A rhombus has diagonals of length 10 cm and 6 cm
  respectively. Find the smaller angle of the
  rhombus.

??
6 cm
10 cm