Pre-Calculus Mathematics 12 5.1 Trigonometric Functions Goal: 1. Measure angle in degrees and radians 2. Find coterminal angles 3. Determine the arc length of a circle Measuring Angles: Angles in Standard Position initial ray/arm on the positive x axis Degree Measure 1/360 of the rotation of a circle is one degree ( ) Θ = 130 Θ = - 100 Θ = 400 Θ = - 530 Coterminal Angles share the same terminal arm Just add or subtract multiples of 360 o Principal Angle - the smallest positive coterminal angle (the angle in standard position) Example 1: Determine a positive and negative coterminal angle of 70 o.
Pre-Calculus Mathematics 12 5.1 Trigonometric Functions Radian Measure a new unit of measuring an angle that is more useful in science and engineering Definition of Radian: A unit of angular measure equal to the angle subtended at the center of a circle by an arc equal in length divided by the radius of the circle Example 1: Convert to radians. i) 120 o ii) 690 o iii) 405 o iv) rotations Example 2: Convert to degrees. i) ii) π iii) iv) rotations Radians measures to memorize: π π π π
Pre-Calculus Mathematics 12 5.1 Trigonometric Functions Arc Length The arc length of a circle is proportional to its central angle and the radius of a circle. The equation for the arc length of a circle ( s = arc length) can easily be found using our knowledge of the circumference. For any circle, C = 2πr Example 3: Determine the arc length of a circle with a radius of 7 cm and a central angle of 130⁰. Example 4: Determine the distance that the tip of the minute hand ( length 10 cm ) travels in 20 minutes. Practice: Page 213 #1-8 (as many as needed)9-16
Pre-Calculus Mathematics 12 5.2 - Trigonometric Functions of an Acute Angle Goal: 1. Explore the six trigonometric ratios Trigonometric Ratios Recall from previous math classes, the trig ratios for an acute angled right triangle. We can also define these in terms of an angle in standard position and a point P( x, y ) And Pythagoras can help us find the radius. Reference angle - the acute angle formed between the terminal arm and the nearest x axis. Eg. 150 o Eg. Reference angle = Reference angle =
Algebraic signs of the Trig Ratios Pre-Calculus Mathematics 12 5.2 - Trigonometric Functions of an Acute Angle The sign of the trig function depends on the quadrant that the function is in. We can easily determine this using the ratios and we will find an easy way to remember this. Trig. Ratios I II III IV sin θ = cos θ = tan θ = Example 1: Determine sin θ if and tan θ > 0. Example 2: Determine sec θ if
Pre-Calculus Mathematics 12 5.2 - Trigonometric Functions of an Acute Angle Example 3: Find the 6 trig ratios if the terminal arm of angle θ contains the point P ( 5, 3 ). Example 4: Given θ in standard position with its terminal arm in the stated quadrant, find the exact values for each trig ratio: in quadrant III. Practice: Page 220 #1 6 (as needed)
Pre-Calculus Mathematics 12 5.3 Trigonometric Functions General & Special Angles Goal: 1. Use special angles to find exact trig values 2. Use quadrantal angles to find exact trig values Warmup: Given θ in standard position with its terminal arm in the stated quadrant, find the exact values for each trig. ratio. Special Triangles as Reference Angles: Recall from previous courses the two special triangles that we can use as a reference. This will allow us to find the exact value of any of the special angles. sin cos tan This allows us to find the exact value of any of the special angles or multiples of the special angles. Just follow this technique, Draw Quadrant(s) terminal of arm(s) in correct quadrant(s) Draw in reference angle(s)/triangle(s) Determine unknown angle(s) / sides Give angle in standard position /give exact value(s) of trig ratio
Pre-Calculus Mathematics 12 5.3 Trigonometric Functions General & Special Angles Example 1: Determine the exact value of Example 2: Determine the exact value of Example 3: Solve. Example 4: Solve. Example 5: Find the exact value of Example 6: Given Find an angle x such that x y,
Pre-Calculus Mathematics 12 5.3 Trigonometric Functions General & Special Angles Quadrantal Angles: or 0, Angles having their terminal side lying along a coordinate axis are quadrantral angles. We can find the exact values of these angles as well. sin = cos = tan = csc = sec = cot = We can use these angles now and add to the angles that we can find the exact values of 2π sin cos tan csc sec cot Example 7: Determine all possible values of x,. Practice: Page 231 #1-10 (as many as needed)
Pre-Calculus Mathematics 12 Exploring the Sine and Cosine Graphs Goal: 1. Determine the graph of sin θ and cos θ Use the following chart to determine the graph of y = sin θ for the given values x = θ (deg) 0 30 45 60 90 120 135 150 180 210 225 240 270 300 315 330 360 x = θ (rad) y = sin θ exact value y = sin θ 2 decimals Plot the points on the graph and connect with a smooth curve to determine the graph of y = sin θ Use the following chart to determine the graph of y = cos θ for the given values x = θ (deg) 0 30 45 60 90 120 135 150 180 210 225 240 270 300 315 330 360 x = θ (rad) y = cos θ exact value y = cos θ 2 decimals Plot the points on the graph and connect with a smooth curve to determine the graph of y = cos θ
Pre-Calculus Mathematics 12 Exploring the Sine and Cosine Graphs Transformation of a Sine Graph Given the function, graph the transformed funciton 3 2 1-1 π 2π 3π 4π -2-3 Properties of a sine and cosine graph Sine Function: y = sin x Cosine Function: y = cos x Both have a period of or Both have an Amplitude = Amplitude = max min 2 Amplitude is ALWAYS Sine curve shifted units to the left becomes the curve
Pre-Calculus Mathematics 12 5.3 Graphing Basic Trigonometric Functions Goal: 1. Transform the graph of sin θ nd θ Comparing: y = sin θ and y = a sin θ Graph,, and In general for y = a sin bx, a controls the amplitude: >1 vertical expansion <1 vertical compression reflection in the x axis Comparing: y = cos θ and y = cos bθ Graph,, and In general, b controls the period: > 1 horizontal compression < 1 horizontal expansion For y = cos bx the new period is: also Each period of a sine or cosine function can be divided into 4 intervals Each interval is a maximum, a minimum, or x intercept.
Graphing: y = a cos bθ and y = a sin bθ Pre-Calculus Mathematics 12 5.3 Graphing Basic Trigonometric Functions Using the value of the amplitude and the value of the period will determine the necessary scale for the graph. The y-scale must include and the x scale must be an n The graph then follows the pattern for the basic sine or cosine function (max /min/x int) Example 1: Sketch the graph over two periods. y = 3 sin 2x Example 2: Sketch the graph over two periods. When transforming ( ) ( ) the c value is just the horizontal translation. When working with periodic functions is called the phase shift. The d value is still referred to as the vertical translation. This is known as the midline of the function. The midline now plays the role that that x-axis did before the transformation. Note: As with any transformation, the coefficient on x must equal 1. If it doesn t, then the coefficient must be factored out of the brackets.
Pre-Calculus Mathematics 12 5.3 Graphing Basic Trigonometric Functions Example 3: Given the function: vertical displacement. y 2 6cos x 2, determine the amplitude, period, phase shift, and 14 Example 4: Given the function: y 3sin 2x 4, determine the amplitude, period, phase shift, 3 and vertical displacement. Example 5: Determine the domain and range of the function: y x 4cos3 1 3 Practice: Page 242 #1-5
Graphing a Sine and Cosine Function To graph a sine or cosine function it must first be in the form: Pre-Calculus Mathematics 12 5.3 Graphing Basic Trigonometric Functions ( ) ( ) or = amplitude ( ) ( ) b also c: phase shift (horizontal translation) d: vertical translation/displacement Graphing steps: 1. Write the function in the form: ( ) ( ) ( ) ( ) 2. Plot Midline: d <--------> 3. Establish min/max from the amplitude: and label on the y-axis 4. Determine period: and determine the interval value: this is the x-scale 5. Establish the new starting point, after phase shift c, of the basic sin/cos graph 6. Graph key points at each interval: maximum/minimum/midline 7. Continue pattern to begin function at x = 0, showing at least one full period Example 3: Graph.
Example 4: Graph. n ( ) Pre-Calculus Mathematics 12 5.3 Graphing Basic Trigonometric Functions Example 5: Graph. ( ) Practice: Page 231 #6-10
Pre-Calculus Mathematics 12 5.5 Applications of Periodic Functions Goal: 1. Use the Unit Circle to graph and solve a trigonometric function 2. Explore other trigonometric functions and their graphs 3. Solve applications involving trigonometric functions The Unit Circle A simple yet powerful way to determine exact values and graph the sine and cosine function is through the use of the unit circle. The unit circle is a circle on the Cartesian plane with a radius equal to 1 (unit length). Using special triangles with a radius 1, we can find all the coordinates of points on the unit circle. From the unit circle, the x-coordinate is equivalent to and the y-coordinate is Example 1: Using the unit circle, determine the exact value of Example 2: Using the unit circle, solve for θ in
Graphing Other Trigonometric Functions Using the graph of sine, determine the reciprocal, the graph of cosecant. Pre-Calculus Mathematics 12 5.5 Applications of Periodic Functions Using the graph of cosine, determine the reciprocal, the graph of secant. Using your graphing calculator, determine the graph of tangent and cotangent
Applications of Trigonometric Functions Pre-Calculus Mathematics 12 5.5 Applications of Periodic Functions Sine or cosine functions can be used to describe periodic or harmonic motion, motion that repeats over a fixed time interval. There are many real life types of motion that can be modeled using a periodic function such as pendulums, springs, Ferris wheels, alternating electrical current (AC), tides, heart beats, annual temperatures and rainfall, radio waves, etc. To solve questions involving periodic motion, we need to first determine the sinusoidal function (either sine or cosine) that models the motion. The easiest way to do this is to first create the graph of the motion. Example 3: The bottom of a Ferris wheel is 2 m above the ground. The wheel rotates once every 20 seconds and has a diameter of 12m. Brandon gets on the Ferris wheel at the bottom. a) Graph the sinusoidal motion of Brandon s height as a function of time for his first minute on the ride. b) Determine sine and cosine equations for the height as a function of time. c) Determine his height above the ground at 12 seconds. d) Determine the total time he is 10.5m or higher above the ground while he is on the ride for 1 minute.
Pre-Calculus Mathematics 12 5.5 Applications of Periodic Functions Example 4: Tides are a periodic rise and fall of water in the ocean. A low tide of 4.2 metres occurs at 4:30 am and a high tide occurs at 11:30 am on the same day. Determine the height of the tide at 1: 30 pm on that same day. Practice: Page 247 #1-12
Pre-Calculus Mathematics 12 Exploring the Sine and Cosine Graphs Goal: 1. Determine the graph of sin θ and cos θ Use the following chart to determine the graph of y = sin θ for the given values x = θ (deg) 0 30 45 60 90 120 135 150 180 210 225 240 270 300 315 330 360 x = θ (rad) y = sin θ exact value y = sin θ 2 decimals Plot the points on the graph and connect with a smooth curve to determine the graph of y = sin θ Use the following chart to determine the graph of y = cos θ for the given values x = θ (deg) 0 30 45 60 90 120 135 150 180 210 225 240 270 300 315 330 360 x = θ (rad) y = cos θ exact value y = cos θ 2 decimals Plot the points on the graph and connect with a smooth curve to determine the graph of y = cos θ
Pre-Calculus Mathematics 12 Exploring the Sine and Cosine Graphs Transformation of a Sine Graph Given the function, graph the transformed funciton Properties of a sine and cosine graph Sine Function: y = sin x Cosine Function: y = cos x Both have a period of or Both have an Amplitude = Amplitude = max min 2 Amplitude is ALWAYS Sine curve shifted units to the left becomes the curve