3.3 Using inverse trigonometric functions to solve trigonometric equations

• What are trigonometric equations primarily concerned with finding?

  Values of the variable
• An example of a sine trigonometric equation is \sin x = \frac{1}{2}
• What is the general form of a cosine trigonometric equation?

  \cos x = b</latex>
• In trigonometric equations, $x$ represents the angle.
• What do inverse trigonometric functions return?

  The angle
• The range of the inverse sine function is \[ - \frac{\pi}{2}, \frac{\pi}{2}]
• Why is $\arcsin(\frac{1}{2}) =$$\frac{\pi}{6}$?


  $\sin(\frac{\pi}{6}) =$$\frac{1}{2}$
• The range of the inverse cosine function is \[0, \pi].
• What are the principal values of inverse trigonometric functions used for?

  Ensuring unique solutions
• The principal value range of $\arcsin(x)$ is \[ - \frac{\pi}{2}, \frac{\pi}{2}]
• If $\arcsin(x) =$$\theta$, then $- \frac{\pi}{2} \le \theta \le \frac{\pi}{2}$.
• Steps to solve trigonometric equations using inverse functions

  1️⃣ Find the initial solution within the principal value range

  2️⃣ Determine other angles where the trigonometric function has the same value

  3️⃣ Add multiples of $2\pi$ (or $\pi$ for tangent) to find all possible solutions
• What is the initial solution for $\sin x =$$\frac{1}{2}$ using inverse sine?


  $\frac{\pi}{6}$
• An example of a cosine trigonometric equation is \cos x = \frac{\sqrt{3}}{2}
• The inverse trigonometric function for sine is denoted as \arcsin(x)
• The principal value range for $\arcsin(x)$ is \[ - \frac{\pi}{2}, \frac{\pi}{2}].
• The principal value range for $\arccos(x)$ is \[0, \pi]
• Extraneous solutions in trigonometric equations arise due to squaring or transforming expressions that change the domain or range.
• Steps to identify and eliminate extraneous solutions in trigonometric equations:

  1️⃣ Solve the equation using algebraic methods

  2️⃣ Substitute each potential solution into the original equation

  3️⃣ Verify if each solution holds true

  4️⃣ Discard solutions that do not satisfy the original equation
• The solutions to $\tan x +$$\sqrt{3} =$$0$ in the range \[0, 2\pi] are \frac{2\pi}{3} and $\frac{5\pi}{3}$.
• What is the first step in solving a trigonometric equation using inverse functions?

  Find initial solution
• The initial solution to $\cos x =$$\frac{\sqrt{3}}{2}$ is \frac{\pi}{6} within its principal value range.
• When solving $\cos x =$$\frac{\sqrt{3}}{2}$, the other valid solution in the range \[0, 2\pi] is $\frac{11\pi}{6}$.
• Trigonometric equations involve trigonometric functions and ask for the values of the variable that make the equation true.
• What are the solutions to the equation $\cos x =$$\frac{\sqrt{3}}{2}$ within the interval $0 \le x \le 2\pi$?


  $\frac{\pi}{6}$ and $\frac{11\pi}{6}$
• Adding multiples of $2\pi$ is necessary to find solutions within the interval $0 \le x \le 2\pi$.


  False
• Trigonometric equations involve trigonometric functions such as sine, cosine, or tangent
• What is the general form of a cosine equation in trigonometry?

  $\cos x =$$b$
• What is the value of $x$ in the equation $\tan x =$$1$?


  $\frac{\pi}{4}$
• The values of $a$, $b$, and $c$ in trigonometric equations must be real numbers.
• Inverse trigonometric functions return the angle whose trigonometric value is a given number.
• What is the notation for inverse sine?

  arcsin(x)
• The principal value range for arcsin(x) is $[ - π / 2, π / 2]$.
• Inverse trigonometric functions return the angle whose trigonometric value is a given number.
• What is another notation for inverse sine?

  $\sin^{ - 1}(x)$
• arcsin(1/2) = π/6 because sin(π/6) = 1/2.
• Why are principal values used in inverse trigonometric functions?

  To ensure unique solutions
• The principal values of inverse trigonometric functions are restricted ranges of angles for which the inverse functions are defined.
• What is the principal value range for arcsin(x)?

  $[ - π / 2, π / 2]$
• What is the principal value range for arccos(x)?

  [0, π]</latex>