Compound Angle Formula

• What chapter discusses the double angle and addition formulae?

  Chapter 11
• What is the first addition formula for cosine?

  cos(a + b) = cos(a)cos(b) - sin(a)sin(b)
• What is the second addition formula for cosine?

  cos(a - b) = cos(a)cos(b) + sin(a)sin(b)
• What must you know by heart for this chapter?

  The sine, cosine, and tangent graphs, and the exact value triangles or table
• How do you expand cos(g + h)?

  • cos(g + h) = cos(g)cos(h) - sin(g)sin(h)
• How do you expand cos(n - 35)?

  • cos(n - 35) = cos(n)cos(35) + sin(n)sin(35)
• What is the goal when simplifying cos(170)cos(70) - sin(170)sin(70)?

  To recognize it as cos(170 + 70)
• What is cos(240) when simplified from cos(170 + 70)?

  -cos(60)
• What is the value of cos(60)?

  \(\frac{1}{2}\)
• How do you simplify cos(\(\frac{\pi}{2}\))cos(\(\frac{\pi}{3}\)) + sin(\(\frac{\pi}{2}\))sin(\(\frac{\pi}{3}\))?


  Recognize it as cos(\(\frac{\pi}{2} - \frac{\pi}{3}\))
• What is \(\frac{\pi}{2} - \frac{\pi}{3}\) in terms of a common denominator?

  \(\frac{\pi}{6}\)
• What is the cosine of \(\frac{\pi}{6}\)?

  \(\frac{\sqrt{3}}{2}\)
• What is the sine of \(\frac{\pi}{6}\)?

  \(\frac{1}{2}\)
• How do you simplify cos(x - y) - cos(x + y)?

  1. Expand cos(x - y) = cos(x)cos(y) + sin(x)sin(y)
  2. Expand cos(x + y) = cos(x)cos(y) - sin(x)sin(y)
  3. Combine and simplify to get 2sin(x)sin(y)
• What is the result of expanding cos(x + \(\frac{\pi}{6}\))?


  cos(x)cos(\(\frac{\pi}{6}\)) - sin(x)sin(\(\frac{\pi}{6}\))
• What is the value of cos(\(\frac{\pi}{6}\))?

  \(\frac{\sqrt{3}}{2}\)
• What is the value of sin(\(\frac{\pi}{6}\))?

  \(\frac{1}{2}\)
• What is the value of cos(\(\pi\))?

  -1
• What is the value of sin(\(\pi\))?

  0
• How do you find sin(a) and sin(b) given cos(a) and cos(b)?

  Use triangles to find the opposite sides based on the adjacent and hypotenuse values.
• If cos(a) = \(\frac{12}{13}\), what is sin(a)?

  \(\frac{5}{13}\)
• If cos(b) = \(\frac{3}{5}\), what is sin(b)?


  \(\frac{4}{5}\)
• How do you calculate cos(a - b)?

  Use the formula cos(a)cos(b) + sin(a)sin(b)
• What is the result of cos(a)cos(b) + sin(a)sin(b) when substituting the values for a and b?


  \(\frac{36}{65} + \frac{20}{65} = \frac{56}{65}\)
• How can you express cos(15 degrees) using known angles?

  cos(15 degrees) = cos(45 degrees - 30 degrees)
• What is the expanded form of cos(45 degrees - 30 degrees)?

  cos(45)cos(30) + sin(45)sin(30)
• What is the cosine of 45 degrees?

  \(\frac{1}{\sqrt{2}}\)
• What is the sine of 45 degrees?

  \(\frac{1}{\sqrt{2}}\)
• What is the cosine of 30 degrees?

  \(\frac{\sqrt{3}}{2}\)
• What is the sine of 30 degrees?

  \(\frac{1}{2}\)
• How do you rationalize the denominator of a fraction?

  Multiply the numerator and denominator by the same radical to eliminate the radical in the denominator.
• What is the result of rationalizing the denominator of \(\frac{root{3} + 1}{2\sqrt{2}}\)?

  \(\frac{\sqrt{6} + \sqrt{2}}{4}\)
• What is the final answer for cos(15 degrees) when simplified?

  \(\frac{\sqrt{6} + \sqrt{2}}{4}\)
• What should you do if you encounter problems while practicing these examples?

  Ask for help.
• What is the addition formula for sine?

  Sine a plus b can be expanded to sine a cos b plus cos a sine b.
• How can sine a minus b be expressed using the addition formula?

  Sine a minus b can be written as sine a cos b minus cos a sine b.
• What do you need to know for the exam regarding sine, cosine, and tangent?

  You need to know their exact values and their graphs off by heart.
• What are the steps to expand sine u plus v?

  • Replace a with u and b with v in the formula.
  • The expansion becomes sine u cos v plus cos u sine v.
• How do you expand sine p minus q?

  • Replace a with p and b with q in the formula.
  • The expansion becomes sine p cos q minus cos p sine q.
• How would you find the exact value of sine 105 degrees using the addition formula?

  Write it as sine of 60 degrees plus 45 degrees and then expand using the addition formula.