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Trigonometric Functions of Angles

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Trigonometric Functions of Angles Trigonometric Functions of Angles Source: D.Lippman, M.Rasmussen (2012) Precalculus: An Investigation of Functions (Edition 1.3) Circles : The Pythagorean Theorem the sum of the squares of the legs of a right triangle will equal the square of the hypotenuse of the triangle. ퟐ ퟐ ퟐ 풂 + 풃 = 풄 Distance between two 풙ퟏ, 풚ퟏ 풂풏풅 풙ퟐ, 풚ퟐ 2 2 points 풅풊풔풕 = (푥2 − 푥1) + (푦2 − 푦1) Radius Equation 풓 = (풙 − 풉)ퟐ + (풚 − 풌)ퟐ Circle Equation 풙 − 풉 ퟐ + 풚 − 풉 ퟐ = 풓ퟐ Angles: Measure of an angle is a measurement between two intersecting lines A degree is a measurement of angle. Coterminal Angles After completing their full rotation based on the given angle, two angles are coterminal if they terminate in the same position Arclength Arclength is the length of an arc, s, along a circle of radius r subtended (drawn out) by an angle 휃 Arclength on a Circle The length of an arc, s, along a circle of radius r subtended by angle θ in radians is 푠 = 푟휃 Area of a Sector of a circle with radius r subtended by an angle θ , measured in radians, is 1 2 = 휃푟 2 Radians The radian measure of an angle is the ratio of the length of the circular arc subtended by the angle to the radius of the circle. 푠 (푙푒푛푔푡푕 표푓 푎푛 푎푟푐) 푟푎푑푖푎푛 푚푒푎푠푢푟푒 = 푟 ( 푟푎푑푖푢푠) 휋 180 1 푑푒푔푟푒푒 = or 1 푟푎푑푖푎푛 = 180 휋 Angular velocity(ω) point moves along a circle of radius r, can be found as the angular rotation θ per unit time, t. 휃 휔 = 푡 Linear Velocity( v) of the point can be found as the distance travelled, arclength s, per unit time, t. 푠 푣 = 푡 When the angular velocity is measured in radians per unit time, linear Relationship Between velocity and angular velocity are related by the equation Linear and Angular v r = ω Velocity 푦 The sine function sin 휃 = 푟 푥 The cosine function cos 휃 = 푟 Domain of Sine and Cosine is all real numbers, ( −∞, ∞) The range of sine and is the interval [-1, 1] cosine Graph sinusoidal (the shape of the graph begins repeating after 2π) The slope of the sine curve is given by the cosine curve. The slope of the cosine curve follows the negative of the sine curve. Coordinates of the point 푥 = 푟푐표푠(휃) (x, y) 푦 = 푟푠푖푛(휃) 푐표푠2 휃 + 푠푖푛2 휃 = 1 The Pythagorean Identity Reference Angle is the size of the smallest angle to the horizontal axis(between 0 and 90 degrees) 푦 sin ⁡(휃) The tangent function: tan 휃 = = 푥 cos ⁡(휃) 푟 1 The secant function: sec 휃 = = 푥 cos ⁡(휃) 푟 1 The cosecant function: csc 휃 = = 푦 sin ⁡(휃) 푥 cos (휃) The cotangent function: cot 휃 = = 푦 sin ⁡(휃) Identities 1 + cot⁡(훼) = sin 훼 + cos⁡(훼) csc⁡(훼) 푐표푠2(휃) = 1 − sin⁡(휃) 1 + sin⁡(휃) 푠푖푛2 휃 = 1 − 푐표푠2(휃) 푐표푠2 휃 = 1 − 푠푖푛2(휃) 휋 Cofunction Identities cos 휃 = sin⁡( − 휃) 2 휋 sin 휃 = cos⁡( − 휃) 2 The Sum and Difference cos 훼 − 훽 = cos 훼 cos 훽 + sin 훼 sin⁡(훽) Identities cos 훼 + 훽 = cos 훼 cos 훽 − sin 훼 sin⁡(훽) sin 훼 + 훽 = sin 훼 cos 훽 + cos 훼 sin⁡(훽) sin 훼 − 훽 = sin 훼 cos 훽 − cos 훼 sin⁡(훽) The Sum to Product 푢 + 푣 푢 − 푣 sin 푢 + sin 푣 = 2sin⁡( )cos⁡( ) Identities 2 2 푢 − 푣 푢 + 푣 sin 푢 − sin 푣 = 2 sin cos⁡( ) 2 2 푢 + 푣 푢 + 푣 cos 푢 + cos 푣 = 2cos⁡( )cos⁡( ) 2 2 푢 + 푣 푢 − 푣 cos 푢 − cos 푣 = −2sin⁡( )sin⁡( ) 2 2 The double angle sin 2훼 = 2 sin 훼 cos⁡(훼) Identities cos 2훼 = 푐표푠2 훼 − 푠푖푛2 훼 = 1 − 2푠푖푛2 훼 = 2푐표푠2 훼 − 1 Power Reduction Identities cos 2훼 + 1 푐표푠2 훼 = 2 1 − cos⁡(2훼) 푠푖푛2 훼 = 2 Half-Angle Identities 휃 cos 휃 + 1 cos = ± 2 2 휃 1 − cos⁡(휃) sin = ± 2 2 푕 Law of Sines sin 훼 = 푏 푕 sin 훽 = 푎 sin ⁡(훼) sin ⁡(훽) sin ⁡(훾) = = 푎 푏 푐 푥 Law of Cosines cos 훼 = or 푥 = 푐 ∗ cos⁡(훼) 푐 푎2 = 푐2 + 푏2 − 2푏푐 ∗ cos⁡(훼) 푏2 = 푎2 + 푐2 − 2푎푐 ∗ cos⁡(훽) 푐2 = 푎2 + 푏2 − 2푎푏 ∗ cos⁡(훾) Law of Tangents 1 푎 − 푏 푡푎푛 (훼 − 훽) = 2 푎 + 푏 1 푡푎푛 2 (훼 + 훽) 1 푏 − 푐 푡푎푛 (훽 − 훾) = 2 1 푏 + 푐 푡푎푛 (훽 + 훾) 2 1 푎 − 푐 푡푎푛 (훼 − 훾) = 2 1 푎 + 푐 푡푎푛 (훼 + 훾) 2 .
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