The Wave Function

• What is the expansion of $k \cos(x - a)$ using the addition formula?


  • $k(\cos x \cos a + \sin x \sin a)$
• What should you do in Step 2 of solving trigonometric equations with the wave function?

  Equate the coefficients of $a \cos x +$$b \sin x$ with the correct trigonometric function, ensuring the correct coefficient.
• Why do you square and sum the coefficients in Step 3?

  To calculate $k$ and eliminate $\sin a$ and $\cos a.$
• How do you calculate $a$ in Step 4?


  By using $\sin a$ and $\cos a$ to find $\tan a.$
• What tool can be used to determine the quadrant of $\tan a?$


  • Draw a CAST diagram
  • Tick which quadrants are positive or negative
• What are the two main steps to solve trigonometric equations using the wave function?

  1. Express in form $k \cos(x + a)$ or $k \sin(x + a).$
  2. Solve.
• In what form should you express trigonometric equations to solve using the wave function?

  • $k \cos(x ± \alpha)$ or $k \sin(x ± \alpha)$
• How would you sketch the graph of $y =$$3 \sin(x - 45)°$ in the interval $0 ≤ x ≤ 360?$


  Steps:

  1. Draw the wave on an x-axis and mark the roots.
  2. Draw a y-axis.
  3. Fill in the rest of the information and add/remove sections to fit the required range.
• How would you sketch the graph of $y =$$3 \sin(x - 45)°$ in the interval $0 ≤ x ≤ 360?$


  Steps:

  1. Draw the wave on an x-axis and mark the roots.
  2. Draw a y-axis.
  3. Fill in the rest of the information and add/remove sections to fit the required range.
• What are the forms of the equations for sketching the graph of sine and cosine functions?

  • $y =$$k \sin(x ± \alpha)$
  • $y =$$k \cos(x ± \alpha)$
• How does the parameter $\alpha$ affect the graph of a trigonometric function?


  • Shifts the graph horizontally to the right by $\alpha$ radians.
• What are maximum and minimum values in trigonometric functions?

  • The highest or lowest values that a trigonometric function reaches within a given range.
• How can the maximum or minimum value of a trigonometric function be determined?

  • By determining the value of $k$ in the function $y =$$k \sin(x ± \alpha)$ or $y =$$k \cos(x ± \alpha).$
• To find the x-coordinate where the maximum or minimum value occurs, what equation should you use?

  Equate the function to the maximum or minimum value, then solve.