1. The Pythagorean Theorem

Transcription

1 . The Pythagorean Theorem The Pythagorean theorem states that in any right triangle, the sum of the squares of the side lengths is the square of the hypotenuse length. c 2 = a 2 b 2 This theorem can be interpreted geometrically to mean that the two squares built on the legs together fit exactly into the square on the hypotenuse. You should be familiar with at least one proof of the theorem. 2. Square Roots and Their Properties You should recognize that certain square roots can be represented as sides of right triangles. 3. Defining Trigonometric Ratios Parallel lines have equal slopes. Perpendicular lines have slopes that are negative reciprocals. In the triangles, sin B = opposite hypotenuse = AC AB cos B = adjacent hypotenuse = BC AB tan B = opposite adjacent = AC BC

2 4. Trigonometric Equations A unit circle is a circle centered at the origin of a coordinate plane and having a radius of one. Consider the point P,0 on the unit circle. Under carious rotations represented by, P has different images on the unit circle. The coordiantes of these new image points can be defined by the following mapping rule. R,0 cos,sin Translated, this means that when the point P,0 is rotated by a magnitude of about the origin, the cosine of and the sine of are the x - and y-coordinates of the image point, respectively. Special Rotation on the Unit Circle R 45 O,0 cos 45 O,sin 45 O 2 2, 2 2 R 30 O,0 cos30 O,sin 30 O 3 2, 2 R 60 O,0 cos60 O,sin 60 O 2, Trigonometric Identities Trigonometric equations can be proven to be identities by using various forms of equivalent relationships to simplify the equation and show that the left side of the equation is the same as the right side. An understanding of trigonometric identities is very important when examining the relationships between various trigonometric expressions. Reciprocal Identities csc x = sec x = sin x cos x cot x = tan x Quotient Identities tan x = sin x cot x = cos x cos x sin x

3 Pythagorean Identities sin 2 x cos 2 x = tan 2 x = sec 2 x cot 2 x = csc 2 x Compound Angle Identities sin a±b = sin a cos b ± cos a sin b cos a±b = cos a cos b sin a sin b tan a ± tan b tan a±b = tan a tan b Double Angle Identities sin 2a = 2 sin a cos a cos 2a = cos 2 a sin 2 a = 2 sin 2 a = 2 cos 2 a tan 2a = 2 tan a tan 2 a in the double angle identity for tangent, the value of a is restricted. The denominator of the right-hand side cannot be zero: tan 2 a. This means that a 4 k 2, k Z. This could also be derived by considering the restrictions imposed by tangent on the value of 2a. The numerator of the right-hand side gives a 2 k,k Z. 6. Law of Sines and Law of Cosines The law of sines and law of cosines are used when parts of an oblique triangle are unknown. Law of Sines sin A a = sin B b Law of Cosines = sin C c c 2 = a 2 b 2 2a b cosc a 2 = b 2 c 2 2bc cos A b 2 = a 2 c 2 2a c cos B

4 7. Inverse Trigonometric Functions The inverse trigonometric functions can be used to calculate a radian measure given the arc length intercepted. The inverse relations y = arcsin x, y = arccos x, and y = arctan x are relations in which y is the radian measure of the intercepted arc x. When working with degrees, the inverse relations are y = sin x, y = cos x, and y = tan x, in which y is an angle measure in degrees whose sine, cosine, or tangent is x. A calculator cannot store the infinite number of angles or radian measures corresponding to any sine, cosine, or tangent value, so the calculator stores only one corresponding angle or radian measure. These restricted values give what are called the principal inverse functions, which are denoted using a capital letter; for example, Sin x. The domain and restricted range of each principal inverse function are as follows: y = Sin x: D = {x x, x R}; R = {y 2 y 2, y R} y = Cos x: D = {x x, x R}; R = {y 0 y, y R} y = Tan x : D = {x x R }; R = {y 2 y 2, y R} 8. Circular and Periodic Motion Radian measure can be used to express angle measure in relation to the arc intercepted by the angle. There are 2 radians in a circle. Periodic functions, such as sine and cosine, can be expressed with the radian measure as the domain of the function. Since 2 radians is equivalent to 360 O, it is easy to convert between degrees and radian measure. The length of an arc intercepted by a central angle can be calculated using the formulas s=r, where s is arc length, r is the radius of the circle, and is the central angle expressed in radians. The tangent function is another periodic function, but it behaves differently from sine or cosine. Ther period of the tangent function is. Since tan = sin cos, tan is undefined where cos = 0, namely, at = 2 k, k Z. Vertical asymptotes occur at these values of. The value of tan approaches ± as approaches one of these asymptotic values. The odd-even and cofunction properties of trigonometric function provide the following

5 relations: cos x = cos x sin x = sin x tan x = tan x cos x 2 = sin x sin x 2 = cos x These relations can be used to simplify trigonometric identities and equations. 9. Graphing Trigonometric Functions Many real-life relations can be modeled using transformations of the periodic finctions sine, cosine, or tangent. The relations involving sine or cosine can be expressed in the form y D = sin B x C or y D = cos B x C A A where A is the amplitude, B is the number of cycles in 2 radians, C is the phase shift, and y = D is the equation of the sinusoidal axis. The model can be used to make predictions and perform calculations based on the relation. The tangent function can be transformed like other periodic functions. Transformations of y = tan x can be expressed as y D = tan B x C A where A is the vertical stretch of the function y = tan x, B is the number of cycles in radians of the new function, C is the phase shift, and D is the vertical translation. The reciprocal functions y = csc x = sin x, y = sec x = cos x, and y = cot x = are also periodic and exhibit asymptotic behaviour like y = tan x. tan x The reciprocal functions can also be transformed in the same way as sine, cosine, and tangent. The reciprocal functions also exhibit odd-even and cofunction properties. 0. The General Rotational Matrix cos sin sin cos The general rotational matrix can be used to rotate points, figures, or pictures on a plane about the origin. Coordinates x, y, expressed in vector form as x, can be rotated by y multiplying the vector by the general rotational matrix.