Trigonometry by mstfdemirdag

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Trigonometry by mstfdemirdag

  1. 1. Trigonometry
  2. 2. • Before we begin our study of trigonometry, it will be helpful to review these basic concepts and definitions. • An angle is the union of two rays which have the same initial point. • If a directed angle is measured in a clockwise direction from its initial side then the angle is a negative angle . If the angle is measured in a counterclockwise direction then it is a positive angle.
  3. 3. • We can measure angles using different units of measurement. The most common units are degree and radian . We write ˚ to show a degree measurement: one full circle measures 360°. We write R to show a radian measurement: one full circle measures 2πR. Example: convert the following degree measurements to radian. a) 180˚ b) 90˚ c) 150˚ d) 120˚ e) 45˚ f) 30˚ g) 60˚ Example: convert the following radian measurements to degree. a) 2π/3 b) π/6 c) 5π/3 d) 7π/4 e) 2π f) π/18 g) 5π/36 • If two or more angles in standard position (its vertex is at the origin of the plane and its initial side lies along the positive x-axis.) have coincident terminal sides then they are called coterminal angles . For example, 90° and -270° are coterminal angles. 180° and 180° are also coterminal angles. • Let β be an angle which is greater than 360° or less than 0°. Then α is called the a primary directed angle of β if α is coterminal with β and α ∈ [0°, 360°). In other words, α is the angle between 0° and 360° which is coterminal with β. We can write: β = α ± k · 360° or β = α ± 2kπ .
  4. 4. • The circle whose center lies at the origin of the coordinate plane and whose radius is 1 unit is called the unit circle. • The coordinate axes divide the unit circle into four parts, called quadrants. The quadrants are numbered in a counterclockwise direction. Examples: In which quadrant does each angle lie? a) 75° b) 228° c) 305° d) 740° e) –442° f) 7π/3 g) – 17π/5
  5. 5. BASIC TRIGONOMETRIC RATIOS Example: In a right triangle, Example: In a right triangle, Example: In the figure below, ΔABC is a right triangle. Given that AC = 4, BC = 5 and m(∠ACB) = x, find Example: In the figure, ΔABC is a right triangle. Given that AB = 3, AC = 4 and m(∠ACB) = x, find the six trigonometric ratios for x.
  6. 6. TRIGONOMETRIC IDENTITIES Example: The trigonometric ratios are related to each other by equations called trigonometric identities. sin x ⋅ cot x ⋅ sec x Simplify the followings. 1+ sin x 1+ csc x sin x ·cos x sin x + tan x csc x 1+ sin x cos x cos x 1- sin x (sin x + cos x )2 -1 (sin x - cos x )2 -1 cos x 1- sin x + 1- sin x cos x a2 + c2 = b2 Pythagorean Trigonometric Ratios of Some Special Angles theorem sin x tan x = cos x cot x = 1 csc x = sin x 1 sec x = cos x cos x sin x sin2x + cos2x = 1 tan2x + 1 = sec2x cot2x + 1 = csc2x tan x ⋅ cot x = 1 0˚ sin cos tan cot 30˚ 45˚ 60˚ 90˚
  7. 7. Basic Trigonometric Theorems Law of Cosine a2 = b2 + c2 – 2bc ⋅ cos A b2 = a2 + c2 – 2ac ⋅ cos B c2 = a2 + b2 – 2ab ⋅ cos C Law of Sine
  8. 8. Sum and Difference Formulas sin(x + y) = sin x ⋅ cos y + cos x ⋅ sin y sin(x – y) = sin x ⋅ cos y – cos x ⋅ sin y cos(x + y) = cos x ⋅ cos y – sin x ⋅ sin y cos(x – y) = cos x ⋅ cos y + sin x ⋅ sin y tan x + tan y tan( x + y ) = 1- tan x ·tan y tan x - tan y tan( x - y ) = 1+ tan x ·tan y Example: cos 75˚ = ? Example: sin 105˚ = ? Example: tan 75˚ = ? Example: sin 15˚ = ? Example: cos 120˚ = ? Example: sin 135˚ = ?

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