Trigonometry

• What is the sine of 90 degrees?


  1
• What is the sine of 270 degrees?

  -1
• What is the sine of 45 degrees?

  $\frac{1}{\sqrt{2}}$
• What is the sine of 30 degrees?

  $\frac{1}{2}$
• What is the sine of 60 degrees?

  $\frac{\sqrt{3}}{2}$
• What are the two triangles worth remembering for exact values in trigonometry?

  1. Right angle triangle with angles 45°, 45°, and hypotenuse $\sqrt{2}$
  2. Right angle triangle with angles 30°, 60°, and sides 1, $\sqrt{3}$, and 2
• How do you find the sine of 45 degrees using the triangle?

  It is opposite over hypotenuse, which is $\frac{1}{\sqrt{2}}$
• How can you find the sine of 300 degrees using the cast diagram?

  By recognizing that 300 degrees is in the fourth quadrant and corresponds to 60 degrees
• What is the sine of 300 degrees based on the cast diagram?

  -$\frac{\sqrt{3}}{2}$
• What is the relationship between degrees and radians?

  $\pi \text{ radians} =$$180^\circ$
• How do you find the cosine of $\frac{\pi}{6}$?


  By recognizing that $\frac{\pi}{6}$ corresponds to 30 degrees
• What is the tangent of $\frac{3\pi}{4}$?


  -1
• How do you find the sine of 150 degrees using the cast diagram?

  By recognizing that 150 degrees is in the second quadrant and corresponds to 30 degrees
• What is the sine of 150 degrees?

  $\frac{1}{2}$
• How do you find the tangent of 120 degrees using the cast diagram?

  By recognizing that 120 degrees is in the second quadrant and corresponds to 60 degrees
• What is the tangent of 120 degrees?

  -$\sqrt{3}$
• What is the cosine of 45 degrees?

  $\frac{1}{\sqrt{2}}$
• How do you find the cosine of $\frac{7\pi}{4}$?


  By recognizing that $\frac{7\pi}{4}$ corresponds to 315 degrees
• What is the cosine of 315 degrees?

  $\frac{1}{\sqrt{2}}$
• How do you find the tangent of $\frac{2\pi}{3}$?


  By recognizing that $\frac{2\pi}{3}$ corresponds to 120 degrees
• What is the tangent of 120 degrees?

  $- \sqrt{3}$
• How do you find the cosine of 240 degrees?

  By recognizing that 240 degrees is in the third quadrant and corresponds to 60 degrees
• What is the cosine of 240 degrees?

  -$\frac{1}{2}$
• What is the sine of 3 for the range between 0 and 2π?

  No solutions
• What is the range of the sine function?

  Between -1 and 1
• What is the tangent of -2π/3?

  -$\frac{1}{\sqrt{3}}$
• How do you solve the equation sin x = 1/2 for x between 0 and 360 degrees?

  By finding x = 30° and x = 150°
• How do you solve the equation cos x = -1/5?

  By using the inverse cosine function in radians
• What is the first solution for cos x = -1/5 in radians?

  Approximately 1.17 radians
• How do you find additional solutions for cos x = -1/5?

  By using the cast diagram to find angles in the second and third quadrants
• What is the tangent of -5?

  It can take any value from negative to positive infinity
• How do you solve the equation tan x = -5?

  By finding the inverse tangent and using the cast diagram
• What are the two normal answers for tan x = -5 in the range of 0 to 720 degrees?

  Approximately 101.3° and 258.7°
• How do you find additional answers for tan x = -5 in the range of 0 to 720 degrees?

  By adding multiples of 360 to the normal answers
• What is the period of the sine function when dealing with multiple angles?

  • The period is 360° for sin x
  • The period is 180° for sin 2x
  • The period is 90° for cos 4x
• What is the first step in solving the equation \(2 \sin 2x - 1 = 0\)?

  Rearranging it to make \(\sin 2x\) the subject.
• What is the rearranged form of the equation \(2 \sin 2x - 1 = 0\)?

  \(\sin 2x = \frac{1}{2}\)
• What is the inverse sine of \(\frac{1}{2}\)?

  It is \(30^\circ\).
• Why does \(2x\) equal \(30^\circ\) not directly imply that \(x = 30^\circ\)?

  Because \(2x\) must be halved to find \(x\).
• What are the two angles derived from the equation \(\sin 2x = \frac{1}{2}\) using the CAST diagram?

  They are \(30^\circ\) and \(150^\circ\).