뒤로Precalculus Study Guide: Trigonometric Functions and Graphs
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Q1. Sketch the graphs of the six trigonometric functions and identify their key features.
Background
Topic: Trigonometric Functions and Their Graphs
This question is testing your understanding of the basic shapes and properties of the six trigonometric functions: sine, cosine, tangent, cotangent, secant, and cosecant. You are expected to recognize their graphs and identify important features such as period, amplitude, asymptotes, and intercepts.
Key Terms and Formulas
Period: The length of one complete cycle of the function.
Amplitude: The maximum distance from the midline to the peak (for sine and cosine).
Asymptotes: Lines where the function is undefined (for tangent, cotangent, secant, and cosecant).
Basic formulas:
Step-by-Step Guidance
Recall the general shape of each trigonometric function. For example, sine and cosine are wave-like, while tangent and cotangent have repeating curves with vertical asymptotes.
Identify the period for each function. For and , the period is . For and , the period is .
Mark the key points on the x-axis, such as $0\frac{\pi}{2}$, $\pi$, $\frac{3\pi}{2} for sine and cosine.
For tangent, cotangent, secant, and cosecant, indicate the locations of vertical asymptotes where the function is undefined.
Sketch the basic shape of each function, making sure to show the correct amplitude, period, and asymptotes, but do not complete the final graph or label all intercepts yet.
Try solving on your own before revealing the answer!
Final Answer:
The six trigonometric functions have the following key features:
Sine (): Wave starting at 0, amplitude 1, period .
Cosine (): Wave starting at 1, amplitude 1, period .
Tangent (): Repeats every , vertical asymptotes at .
Cotangent (): Repeats every , vertical asymptotes at .
Secant (): Reciprocal of cosine, vertical asymptotes where .
Cosecant (): Reciprocal of sine, vertical asymptotes where .
Each graph should show the correct period, amplitude (if applicable), and asymptotes. Refer to your worksheet for the sketched graphs.
Q2. Complete the table of values for sine, cosine, and tangent for key angles, and plot them on the number line.
Background
Topic: Evaluating Trigonometric Functions at Key Angles
This question is testing your ability to find the values of sine, cosine, and tangent for standard angles (such as $0\frac{\pi}{6}$, $\frac{\pi}{4}$, $\frac{\pi}{3}$, $\frac{\pi}{2}$, etc.) and to represent these values on a number line or coordinate axis.
Key Terms and Formulas
Unit Circle: A circle of radius 1 centered at the origin, used to define trigonometric functions for all real numbers.
Key values:
, , , ,
, , , ,
Step-by-Step Guidance
List the key angles in both degrees and radians (e.g., $0, , , or $0$, $\frac{\pi}{6}$, $\frac{\pi}{4}$, $\frac{\pi}{3}$, $\frac{\pi}{2}$).
Recall or use the unit circle to find the sine and cosine values for each angle.
Calculate the tangent value for each angle using .
Fill in the table with these values, but do not complete the last two entries yet.
Plot the values on the number line, marking the correct positions for each trigonometric value, but leave the final placements for the last two angles for you to finish.
Try solving on your own before revealing the answer!
Final Answer:
The completed table should have the following values:
, , , ,
, , , ,
, , , , is undefined
On the number line, plot each value at its correct position. Tangent at is undefined, so mark this appropriately.
