right angled trigonometry

Right Angled Trigonometry

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

Hypotenuse

Adja

cent

Opposite

The _________ is always the longest side of the triangle.

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

hypotenuse

opposite

adjacent

Trigonometric RatiosWe define the 3

trigonometric ratios in terms of fractions

of sides of right angled triangles.

Hypotenuse

(HYP)

Adja

cent

(ADJ

)

Opposite (OPP)

SohCahToaSine equals Opposite over HypotenuseCosine equals Adjacent over HypotenuseTangent equals Opposite over Adjacent

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

65 a

x

32

bx

YOU DO:Given the triangle, write all the ratios that could be used to find x by connecting the angle and sides given.56

d

x

c

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

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

61 9.6 cm

x

HYP

OPP

ADJ

Steps to finding the missing sides of a right triangle using trigonometric ratios:

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

3. Set up the equation

61 9.6 cm

x

HYP

OPP

ADJ

Steps to finding the missing sides of a right triangle using trigonometric ratios:

4.Solve the equation to find the unknown.

61 9.6 cm

x

HYP

OPP

ADJ

Practice Together:Find, to 2 decimal places, the unknown length in the triangle.

41

x m7.8 m

YOU DO:Find, to 1 decimal place, all the unknown angles and sides in the triangle.

a m

14.6 m

63

b m

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

5.92 kmHYP

OPP

ADJ

2.67 km

Steps to finding the missing angle of a right triangle using trigonometric ratios:

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

3. Set up the equation

5.92 kmHYP

OPP

ADJ

2.67 km

Steps to finding the missing angle of a right triangle using trigonometric ratios:

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

5.92 kmHYP

OPP

ADJ

2.67 km

Practice Together:Find, to one decimal place, the unknown angle in the triangle.

3.1 km 2.1

km

YOU DO:Find, to 1 decimal place, the unknown angle in the given triangle.

7 m

4 m

Practice: Isosceles Triangles Using what we already know about right

angles in isosceles triangles find the unknown side.

10 cm

x cm

67

YOU DO: Isosceles Triangles Find the unknown angle of the isosceles

triangle using what you already know about right angles in isosceles triangles.

8.3 m

5.2 m

Practice: Circle Problems Use what you already know about right

angles in circle problems to find the unknown angle.

6 cm

10 cm

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

Practice: Other Figures (Trapezoid) Find x given:

10 cm

x cm65 48

YOU DO: Other Figures (Rhombus) A rhombus has diagonals of length 10

cm and 6 cm respectively. Find the smaller angle of the rhombus.

10 cm

6 cm