1.5 Trigonometry

• What is the first step in solving trigonometric equations?

  Isolate the trig function
• To solve trigonometric equations, you first need to isolate the trigonometric function
• The reference angle for $\sin{\theta} =$$\frac{1}{2}$ is $\frac{\pi}{6}$.
• In which quadrants is sine positive?

  First and second
• What is the Pythagorean identity relating sine and cosine?

  $\sin^{2}{\theta} +$$\cos^{2}{\theta} =$$1$
• The Pythagorean identity $1 +$$\tan^{2}{\theta}$ is equal to $\sec^{2}{\theta}$
• $\sec{\theta} =$$\frac{1}{\cos{\theta}}$ is a reciprocal identity.
• Match the trigonometric function with its quotient identity:

  $\tan{\theta}$ ↔️ $\frac{\sin{\theta}}{\cos{\theta}}$

  $\cot{\theta}$ ↔️ $\frac{\cos{\theta}}{\sin{\theta}}$
• What is the reciprocal identity for $\csc{\theta}$?


  $\frac{1}{\sin{\theta}}$
• The general solution for $\sin{\theta}$ is $\theta =$$\alpha +$$2n\pi$ or $\theta =$$\pi - \alpha +$$2n\pi$, where $n$ is an integer
• The reference angle for \sin{\theta} = \frac{1}{2}</latex> is $\frac{\pi}{6}$.
• What is the general solution for $\tan{\theta}$?


  $\theta =$$\alpha +$$n\pi$
• The general solution for $\sin{\theta} =$$\frac{1}{2}$ includes the term 2n\pi
• Steps to solve trigonometric equations

  1️⃣ Isolate the trig function

  2️⃣ Find the reference angle

  3️⃣ Determine the quadrants

  4️⃣ Express the general solutions
• What is the reference angle for $\cos{\theta} =$$\frac{1}{2}$?


  $\frac{\pi}{3}$
• Cosine is positive in the first and fourth quadrants.
• What are the solutions for $2 \cos{\theta} - 1 =$$0$ in $[0, 2\pi]$?


  $\frac{\pi}{3}$ and $\frac{5\pi}{3}$
• The range of $\arccos{x}$ is [0, \pi]
• What trigonometric function is used to find height when given angle and hypotenuse?

  \sin{\theta}</latex>
• What formula is used to find distance given height and angle using tangent?

  $d =$$\frac{h}{\tan{\theta}}$
• The formula to find the hypotenuse given angle and height is $d =$$\frac{h}{\sin{\theta}}$.