05: Trigonometry and Trigonometric Ratios

Trigonometry: the study of how sides and angles of a triangle are related to each other

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The Right Angle Triangle

• A right angle triangle has a 90º angle
• The side opposing the 90º angle is called the hypotenuse
• the other two sides are called the legs

Labelling Triangles

• The side opposite an angle and vertex becomes the lowercase letter of that angle and vertex

Pythagorean theorem

a² + b² = c²

leg² + leg² = hypotenuse²

• used only for side lengths, cannot be used for angles

Trigonometric Ratios

• Secondary trig ratios are primary trig ratios, flipped

cscθ = 1/sinθ

secθ = 1/cosθ

cotθ = 1/tanθ

→ you can solve by using the reciprocal when facing a secondary trig ratio

Angles of Elevation and Depression

Most often seen in word problems

Angle of Elevation

The angle between the line of sight and the horizontal from the eye

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Angle of Depression

The angle when looking down, between the line of sight and the horizontal from the eye

• Because of Z pattern angles, the angle of elevation and depression are actually the same (and are dependent on a question’s phrasing)

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Exact Values

You may not use decimals here, as that is not the true exact value

Special Triangles

• Used to find the exact values for 45º, 45º isosceles triangles and for 60º, 30º triangles (remember: sum of a triangle’s angles must be 180º)

• Remember that you must rationalize your final answer

  

  Trig Ratios, Angles Greater Than 90º

  • Angles in standard position are always measured from the initial arm to the terminal arm, @@counterclockwise@@ (arrow always drawn)
  • Angles measured closckwise are negative

  

The angle, θ, is known as the principal angle (PA)→ its value can fall between 0º and 360º depending on which quadrant the terminal arm is in

The related acute angle (RAA) %%sits between the terminal arm and the x axis%% in any quadrant

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