To access a customizable version of this book as well as other interactive content, visit www.ck12.org. Any reproduction of this book in any format or medium, in whole or in sections, must include the attribution link http://www.ck12.org/saythanks (prominently placed) in addition to the following terms. Teachers may need to download programs from www.timath.com that will implement or assist with the activities.
Any activity that requires a calculator file or application, go to http://www.education.ti.com/calculators/downloads and type the name of the activity or program in the search box. Students will use the graphing calculator to discover the relationship between the trigonometric functions: sine, cosine, and tangent and the ratios of the lengths of the sides of a right triangle. To download Cabri Jr, go to http://www.education.ti.com/calculators/downloads/US/Software/Detail?id=.
To download the calculator file, go to http://www.education.ti.com/calculators/downloads/US/Activities/. For a given triangle, students will find the relationship between the ratios of side lengths and trigonometric functions.
Solutions
Round and Round She Goes
The first problem and the second problem involve students in exploring the relationship between angle measurements and the coordinates of points in the first quadrant. Student Worksheet: Round and Round, She Goes http://www.ck12.org/flexr/chapter/9699, scroll down to the second activity.
Introduction to the Unit Circle
Extending the Pattern
- Trigonometric Patterns
The angle measurements will differ from the exact values in the answer tables due to the Cabri Jr. Make students aware that they may need to use the closest possible angle measure for some values. The questions in the document are starting points for discussions about the patterns in the values of trig functions.
Student Worksheet: Trigonometric Patternshttp://www.ck12.org/flexr/chapter/9699, scroll down to the third activity.
Trigonometric Patterns
Extension – Patterns in reciprocal functions
Sample answer: The values in the first quadrant are repeated in the other quadrants, but have different signs.
C HAPTER
3 TE Circular Functions - TI
Chapter Outline
- Find that Sine
- Temperature graphs
- Hours of Sunlight
- Tides
- Vertical and Phase Shifts
- Amplitude
- Period
- A simple phase shift
- Vertical shift
- Combining transformations
- Bringing it all together
- Getting Triggy With It
- A general trigonometric function
- The effect of the coefficients
- A closer look at amplitude, period, and frequency
Students will use the values in the lists to change the values of the parameters in the trigonometric functions; they will determine the effect each change has on the shape of the graph. Be sure to ask what effect the sign of the aha has on the graph (if negative, the curve is reflected across the x-axis). This problem allows students to review the period of a function of the form f(x) =sin(bx).
Ask them how to obtain the graph ofy=x2+3 from the graph ofy=x2 (translate the graph three units up). Although students may wonder why they can't declare “phase shift=−c,” have them think again about the graph ofy=x2. Ask them how to get the graph from this ofy= (x+3)2(translate three units to the left).
Students will find that the value of the horizontal stretch of this function affects and therefore changes the period of the function. Ask them how to get the graph of y= (x+3)2 from that (translate 3 units to the left). Students systematically investigate the effect of the coefficients on the graph of sine or cosine functions.
Students can use the zero, minimum, and maximum commands (2. [CALC]) to find these characteristics of the graph. Students will examine each coefficient separately to see the effects of each on the graph of the function. The period of a graph of a sine or cosine function is the "width" of one cycle of the wave.
The value of C is called the phase shift, which horizontally shifts the point where a new graph cycle begins. The phase shift of a sine function is the horizontal distance from the y-axis to the first point where the graph intersects the baseline. The phase shift of the cosine function is the horizontal distance from their axis to the peak of the first peak.
4 TE Trigonometric Identities - TI
- Trigonometric Identities
- Prove
- Prove
- Numerical verification
- Verifying trig identities using graphing
- Trig Proofs
- Using the Calculator for verification
- What’s the Difference?
- Exploring the Angle Difference Formula for Cosine
- Applying the Angle Difference Formula
Students should already be familiar with the trigonometric properties of right triangles, similar triangles, and the Pythagorean theorem, as much of the activity is based on these fundamental theorems. In this problem, students will prove two of the basic trigonometric identities using the Pythagorean Theorem. Now students can place the cursor in the center of the hypotenuse and enter the radius.
Students will see a new version of the unit circle with the original right triangle and then a similar triangle drawn to enclose the original triangle. Students are told on the worksheet that the length of the base of the large triangle is 1, the height is Y and the hypotenuse is X. You may need to explain to students that (small M) and (large M) refer to where that part of the relationship comes from and is not actually part of the relationship.
These values are stored in the calculator's lists with the angle measurements in L1, the cosine values in L2, and the sine values in L3. To set the bubble animation for the second graph, students must press ENTER to the left of the equal sign. Students perform trigonometric proofs and use the calculator's graphing capabilities for verification.
The worksheet aims to guide students through the main ideas of the activity while providing more detailed instructions on how to perform specific actions using the calculator tools. The two graphs coincide, so both sides of the equation are equal, as we proved. Angle difference derivatives and cosine sum formulas are optional extensions included in this activity.
The first and second problems engage students in an exploration of the difference formula for cosine. One of the best things about using the unit circle is that their coordinate is always the sine of the angle and the x coordinate is always the cosine of the angle. In this part of the activity, students answer various questions about the angular difference diagram.
Extension #1–Deriving the Angle Difference Formula for Cosine
In other words, just because thex-andy- coordinates only appear in tenths, they really go on for a long time.
Extension #2 –Angle Sum Formula for Cosine
5 TE Inverse Functions and Trigonometric Equations - TI
What’s your Inverse?
This activity is intended to be teacher led, where students will use their graphing calculators to graph the six inverse trig functions. Go over the definition of a limited domain and make sure students understand why they need to find the inverse of trigonometric functions. For the domain and range ofy=sin−1x, review what the domain and range ofy=sinx are.
Conversely, they will be swapped, so the domain is between -1 and 1 and the range must be all real numbers. Therefore, there is a bounded domain ony=sinx so that it can have an inverse (recall thaty=periodic sinxis). Again, review the domain and range ofy=cosx, which are all real numbers for the domain and between -1 and 1 for the range.
Looking at the graph, the domain of the inverse is also between -1 and 1 and the range is between 0 andπ. For secant, cosecant, and cotangent, walk students through how to derive the equation needed to plug into the calculator. Remember that the graph of cotangent differs from tangent by a reflection about the y−axis and a shift of π2.
Because the tangent is an odd function, or tan(−x) = −tanx, its inverse is also odd.
6 TE Triangles and Vectors - TI
- S INE . I T ’ S THE L AW
- Analyzing Heron’s Formula
- The 3, 4, 5 right triangle
- Sine. It’s the Law
- Law of Sines
- Application of the Law of Sines
In this activity, students will use their graphing calculators to determine the relationship between Heron's Formula and the basic area formula. Make sure all students change their window to dimensions on the right before making the graph. Ask students to analyze the domain and range and why there are empty spaces in the graph.
If you enlarge the pupils further, they will see that the nox-intercept and the oneyintercept are at (0, 0). So, from this system of equations, we've figured out what it is to have the correct area, in this case also 6. If we can find the area in more than one way, that will always work as a way to solve for .
This activity is aimed at geometry students and only the simplest case of the Law of Sines is explored. Students will begin this activity by looking at a triangle and investigating the ratio of the sine of one angle to the length of the opposite side. In LAW1.8xv, students are given triangle ABC with angle measures A,B,andC and side measures,b,dc.
Students will collect data in the tables on their accompanying worksheet and be asked what they notice about the last three columns of the table in Question 2. In Problem 2, students are asked to apply what they have learned about the Law of Sines.
Extension –Proof of the Law of Sines
7 TE Polar Equations and Complex Numbers - TI
Polar Necessities
Students will graphically and algebraically find the slope of the tangent line at a point on a polar graph. Finding the area of a region of a polar curve will be determined using the area formula.
Plotting Coordinates & Exploring Polar Graphs
