Appendix D Trigonometry

Transcription

1 Math 151 c Lynch 1 of 8 Appendix D Trigonometry Definition. Angles can be measure in either degree or radians with one complete revolution 360 or 2 rad. Then Example 1. rad = 180 (a) Convert 3 4 into degrees. (b) Convert 30 into radians. Here are some common angles in both degrees and radians: Degrees Radians Angle Subtending an Arc Theorem. Suppose that we have a central angle θ in radians and radius r subtending an arc with length a. Then θ = a r and a = rθ Example 2. (a) If the radius of a circle is 7 in., what angle is subtended by an arc of 4 in.? (b) If a circle has radius 5 cm, what is the length of an arc subtended by a central angle of 5 9?

2 Math 151 c Lynch App. D Trig 2 of 8 Angles Definition. The standard position of an angle occurs when we place its vertex at the origin of a coordinate system and its initial side on the positive x-axis. A positive angle is obtained by rotating the initial side counterclockwise until it coincides with the terminal side. Similarly, negatives angles are obtained by clockwise rotation. Example 3. Graph the normal coordinate plain and label the angle for each axis. Definition. The 2-dimensional or Cartesian plane can be divided into four Quadrants. Example 4. Graph the angles 11 6, 2 3, and 2.7 rad.

3 Math 151 c Lynch App. D Trig 3 of 8 Trigonometric Functions Special Triangles. You should know the sides lengths for the two common triangles: the 4, 4, 2 triangle (45, 45, 90 ) and the 6, 3, 4 triangle (30, 60, 90 ). Definition. For acute angles (between 0 and 2 ), we can define the trig functions at θ using any right triangle containing the angle θ as below. Remember: sohcahtoa sin θ = opp hyp cos θ = adj hyp tan θ = opp adj csc θ = hyp opp sec θ = hyp adj cot θ = adj opp Example 5. Evaluate the six trig functions at the following angles (a) 6 (b) 4 Definition. We can define the trig functions at θ for any value of θ by interpreting θ as an angle in standard position and defining the trig functions using the point (x, y) where the terminal side of the angle intersects a circle at the origin. (Due to similar triangles we may use a circle of any radius).

4 Math 151 c Lynch App. D Trig 4 of 8 If the point is (x, y), and the circle has radius r, then the trig functions are: sin θ = y r cos θ = x r tan θ = y x csc θ = r y sec θ = r cot θ = x y Note. Where is each trig function positive? Remember, All Students Take Calculus Note. You should be familiar with all the common angles and the points on the unit circle (see below), and be able to evaluate the trig functions at those angles. Example 6. Find all six trig functions at the following angles. (a) 4 3 (b) 11 6

5 Math 151 c Lynch App. D Trig 5 of 8 Example 7. If csc x = trig functions. and x is in Quadrant II, find the values of the remaining 5 Example 8. Using the diagram below to find the length of y. Trigonometric Identities You should know the following Trig Identities: csc θ = 1 sin θ Reciprocal Identities tan θ = sin θ cos θ sec θ = 1 cos θ cot θ = cos θ sin θ Pythagorean Identities sin 2 θ + cos 2 θ = 1 tan 2 θ + 1 = sec 2 θ 1 + cot 2 θ = csc 2 θ cot θ = 1 tan θ Even/Odd sin( θ) = sin θ cos( θ) = cos θ Example 9. Find all the values of x in the interval [0, 2] such that 2 sin 2 x 1 = sin x.

6 Math 151 c Lynch App. D Trig 6 of 8 Example 10. If tan x = 7 6 and x is in Quadrant III, find cos 2x. Graphs of Trigonometric Functions You should be familiar with the graphs of the trig functions. y = sin x y = cos x y = sec x y = csc x y = tan x y = cot x

7 Math 151 c Lynch App. D Trig 7 of 8 Lines Definition. The slope of a line, m, is the ratio of the change in y to the change in x. m = rise change in y ( y) = run change in x ( x) The slope of the line through points P 1 (x 1, y 1 ) and P 2 (x 2, y 2 ) is m = y 2 y 1 x 2 x 1 Definition. The equation of a line with slope m that passes through the point (x 1, y 1 ) is y y 1 = m(x x 1 ). This is called the point-slope form of the equation of a line. Example 11. Find the formula for a line that goes through the points (3, 4) and (1, 2). Theorem. A line is in slope-intercept form if it is written as y = mx + b. For this form, we have the following facts: The y-intercept is b so the line intersects the y-axis at the point (0, b). The coefficient m of x is the slope of the line. Theorem. Two lines with slopes m 1 and m 2 are parallel if they have the same slope, i.e., m 1 = m 2. The two lines are perpendicular if the slopes are negative reciprocals, i.e., m 1 = 1 m 2. Example 12. Find an equation of the line that passes through the point (2, 5), and is perpendicular to the line 3x 7y = 4.

8 Math 151 c Lynch App. D Trig 8 of 8 Domain Definition. For a function f(x), the domain of the function is assumed to be any real number where f(x) is defined. Example 13. Find the domain of the function f(x) = x 2 9 2x 2 +9x 18. Rationalization Example 14. Rationalize the numerator and simplify 3(x+h)+7 3x+7 h.