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Trigonometric Ratios in Right TrianglesPowerPoint Presentation

Trigonometric Ratios in Right Triangles

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### Trigonometric Ratios are based on the Concept of Similar Triangles!

b

The ratio is called the Tangent Ratio for angle

The Tangent Ratioc’ a’

b’

If two triangles are similar, then it is also true that:

Tangent q =

Hypotenuse

Side

Opposite

q

q

Side

Adjacent

q

The Tangent RatioThere are a total of six ratios that can be made

with the three sides. Each has a specific name.

Side

Opposite

q

q

Side

Adjacent

q

The Six Trigonometric Ratios(The SOHCAHTOA model)S O H C A H T O A

Side

Opposite

q

q

Side

Adjacent

q

The Six Trigonometric RatiosThe Cosecant, Secant, and Cotangent of q

are the Reciprocals of

the Sine, Cosine,and Tangent of q.

1

Solving a Problem withthe Tangent RatioWe know the angle and the

side adjacent to 60º. We want to

know the opposite side. Use the

tangent ratio:

h = ?

60º

53 ft

Why?

x

q

Trigonometric Functions on a Rectangular Coordinate SystemPick a point on the

terminal ray and drop a perpendicular to the x-axis.

(The Rectangular Coordinate Model)

x

q

Trigonometric Functions on a Rectangular Coordinate SystemPick a point on the

terminal ray and drop a perpendicular to the x-axis.

r

y

x

The adjacent side is x

The opposite side is y

The hypotenuse is labeled r

This is called a

REFERENCE TRIANGLE.

r

y

x

q

x

Trigonometric Values for angles in Quadrants II, III and IVPick a point on the

terminal ray and drop a perpendicular

to the x-axis.

q

x

Trigonometric Values for angles in Quadrants II, III and IVPick a point on the

terminal ray and raisea perpendicular

to the x-axis.

q

x

Trigonometric Values for angles in Quadrants II, III and IVPick a point on the

terminal ray and raise a perpendicular

to the x-axis.

x

y

r

Important! The is

ALWAYS drawn to the x-axis

x

Signs of Trigonometric FunctionsSin (& csc) are

positive in

QII

All are positive in QI

Tan (& cot) are

positive in

QIII

Cos (& sec) are

positive in

QIV

(0, 1)

x

90º

Trigonometric Values for Quadrantal Angles (0º, 90º, 180º and 270º)x = 0

y = 1

r = 1

Pick a point one unit from

the Origin.

r

45 º

1

For Reciprocal Ratios, use the facts:

Trigonometric Ratios may be found by:Using ratios of special triangles

For angles other than 45º, 30º, 60º or Quadrantal angles, you will need to use a calculator. (Set it in Degree Mode for now.)

Acknowledgements

- This presentation was made possible by training and equipment provided by an Access to Technology grant from Merced College.
- Thank you to Marguerite Smith for the model.
- Textbooks consulted were:
- Trigonometry Fourth Edition by Larson & Hostetler
- Analytic Trigonometry with Applications Seventh Edition by Barnett, Ziegler & Byleen

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