Trigonometric Graphs And Equations

Trigonometric Graphs Overview

• Trig graphs are the graphs of $y = \sin x, y = \cos x$, and $y = \tan x$.
• They have repeating (periodic) shapes and symmetries.
• The variable x can be any angle, often between 0° and 360°.

Which of the following is NOT a trigonometric graph?

• $y = \sin x$
• $y = \cos x$
• $y = \tan x$
• $y = x^{2}$

Graph of y = sin x

• The graph of $y = \sin x$ is a wave oscillating between 1 and -1.
• Its period is 360°.
• It passes through (0,0), and key points at multiples of 90°: (0°,0), (90°,1), (180°,0), (270°,-1), (360°,0).

The graph of $y = \sin x$ oscillates between ____ and ____. Its period is ____^{\circ}.

Graph of y = cos x

• The graph of $y = \cos x$ is a wave oscillating between 1 and -1.
• Its period is 360°.
• It has a y-intercept at (0,1), and key points: (0°,1), (90°,0), (180°,-1), (270°,0), (360°,1).

What is the y-intercept of $y = \cos x$?

• (0,0)
• (0,1)
• (0,-1)
• (1,0)

Graph of y = tan x

• The graph of $y = \tan x$ consists of branches repeating every 180°.
• It has vertical asymptotes at $x = 90^{\circ}, 270^{\circ}$, etc.
• It passes through (0,0) and key points: (0°,0), (45°,1), (90°, asymptote), (135°,-1), (180°,0), etc.

Which of these are true about $y = \tan x$?

• Its period is 180°.
• It has asymptotes at $x = 90^{\circ}$ and $x = 270^{\circ}$.
• It oscillates between 1 and -1.

Solving sin x = k

• Use inverse sin on your calculator to find the first solution (acute angle).
• Use the symmetry of the sine graph: if x is a solution, then $180^{\circ} - x$ is another solution.
• For example, $\sin x = 0.5$ gives $x = 30^{\circ}$ and $x = 150^{\circ}$.

Solve $\sin x = 0.5$ for $0^{\circ} \le x \le 360^{\circ}$.

• 30° and 150°
• 30° and 330°
• 60° and 120°
• 150° and 210°

Solving cos x = k

• Use inverse cos to find the first solution.
• Use symmetry: if x is a solution, then $360^{\circ} - x$ is another.
• For example, $\cos x = 0.5$ gives $x = 60^{\circ}$ and $x = 300^{\circ}$.

For $\cos x = 0.5$, the solutions in 0° to 360° are ____^{\circ} and ____^{\circ}.

Solving tan x = k

• Use inverse tan to find the first solution.
• Use periodicity: if x is a solution, then $x + 180^{\circ}$ is another.
• For example, $\tan x = 1$ gives $x = 45^{\circ}$ and $x = 225^{\circ}$.

Solve $\tan x = 1$ for $0^{\circ} \le x \le 360^{\circ}$.

• 45° and 225°
• 45° and 135°
• 45° and 315°
• 135° and 315°

Rearranging Trig Equations

• Some equations need rearranging first, $e.g., 2 \sin x - 1 = 0$ becomes $\sin x = 0.5$.
• Then solve as usual using the appropriate method.

Which of these are correct steps to solve $3 \cos x = 1$?

• Divide both sides by 3 to get $\cos x = \frac{1}{3}$.
• Take inverse cos to get $x =$ cos⁻¹(1/3).
• The second solution is $180^{\circ} - x$.

Negative First Solutions

• Sometimes inverse trig gives a negative angle.
• Extend the graph to the left to find solutions in the given interval.
• Use symmetry or periodicity to find all solutions within 0° to 360°.

If sin⁻¹(-0.5) $= -30^{\circ}$, what are the solutions for $\sin x = -0.5$ in 0° to 360°?

• 210° and 330°
• 150° and 210°
• 30° and 330°
• 210° and 300°

Summary of Key Points

• Sine and cosine have period 360°, tangent has period 180°.
• Sine symmetry: x and 180°-x.
• Cosine symmetry: x and 360°-x.
• Tangent periodicity: x and x+180°.

The period of $y = \tan x$ is ____^{\circ}, while the period of $y = \sin x$ is ____^{\circ}.