Trigonometric Graphs: Parent Functions

Ever wonder why waves, sound, and even seasons repeat? The sine function models these patterns perfectly! The parent function y = sin x is the foundation of trig graphs with some key characteristics:

The sine function has a domain of all real numbers (-∞,∞) and a range of $-1,1$. Its graph creates a wave that increases between 270° and 450° and decreases between 90° and 270°. The center line runs along y = 0, which is where the graph crosses as it oscillates up and down.

Amplitude is how "tall" the wave gets from the center line. For y = A sin x, the amplitude is |A|, meaning it's always positive. For example, in y = -4 sin x, the amplitude is |-4| = 4, and the range becomes $-4,4$. The negative sign causes a vertical reflection over the x-axis.

💡 The distance from maximum to minimum on a sine graph is double the amplitude! This is why the height of the graph equals 2 times the amplitude.

Vertical Translations and Center Lines

When you move a wave up or down, everything shifts! In y = A sin x + D, the D value pushes the entire graph vertically.

The center line shifts from y = 0 to y = D, and the new range becomes $−A+D, A+D$. Let's see this in action: for y = 2 sin x + 3, the amplitude is 2 and the range becomes $-2+3, 2+3$ = $1, 5$.

Think of it like moving the entire wave up 3 units, so the center line is now at y = 3. All the key points shift upward, but the wave still has the same shape and amplitude.

This is different from horizontal shifts (which we'll see next), so be careful not to confuse y = A sin x + D with y = A sin$x + D$!

Phase Shifts and Periods

Shifting a trig graph horizontally creates a phase shift. In y = sin$x - C$, the graph shifts C units right when C is positive. This doesn't change the domain (-∞,∞) or range $-A+D, A+D$, but it does move all the key points.

For example, in y = sin$x - 45°$, there's a 45° phase shift to the right, meaning the center line points move from (0,0), (180°,0), and (360°,0) to (45°,0), (225°,0), and (405°,0).

The period of a trig function determines how long it takes to complete one full cycle. For y = sin(Bx), the period is 360°/B. The B value essentially stretches or compresses the graph horizontally.

🔑 Remember this formula! The period of sine and cosine is always 360°/B where B is the coefficient of x inside the function.

Combining All Transformations

When all transformations happen at once, you get y = A sin$B(x-C)$ + D. This might look complicated, but let's break it down with an example: y = 3 sin$3x - 135°$ - 1.

First, rewrite it as y = 3 sin$3(x - 45°)$ - 1. Now we can identify each transformation:

• Amplitude: 3
• Period: 360°/3 = 120°
• Phase shift: 45° right
• Vertical shift: Down 1 unit $center line at y = -1$

The domain remains (-∞,∞), but the range becomes $-3-1, 3-1$ or $-4,2$.

The center line points have moved to (45°,-1), (45°+60°,-1), and (45°+120°,-1), or simplified: (45°,-1), (105°,-1), and (165°,-1).

By breaking down these transformations step-by-step, you can graph even the most complex trig functions!

Right Triangle Trigonometry

When working with right triangles, knowing which side is which is half the battle! The hypotenuse is always the side opposite the 90° angle. The opposite and adjacent sides depend on which angle you're focusing on.

Here's the key to keeping them straight:

• If using angle 1, the opposite is side a, and adjacent is side b
• If using angle 2, the opposite is side b, and adjacent is side a
• The hypotenuse is always side c (across from the 90° angle)

The basic trig ratios connect these sides:

• Sine = opposite/hypotenuse (use when you have the hypotenuse and opposite side)
• Cosine = adjacent/hypotenuse (use when you have the hypotenuse and adjacent side)
• Tangent = opposite/adjacent (use when you have both legs but not the hypotenuse)

🧠 Think about which angle you're using before setting up your ratio! The opposite and adjacent sides will change depending on your reference angle.

Solving Right Triangle Problems

Right triangles pop up everywhere in real life! To solve them, identify what you know and pick the right trig ratio.

Example 1: To find side b when a = 4 and angle 1 = 25°, use tangent. Since we know angle 1, side a (opposite), and want side b (adjacent): tan 25° = 4/b b = 4/tan 25° = 8.578

Example 2: To find side a when side b = 6 and hypotenuse c = 12, use cosine. Since we know angle 2, hypotenuse c, and want side a (adjacent): cos angle 2 = a/12 a = 12 × cos angle 2 = 1.877

Example 3: To find angle 2 when b = 14 and c = 18, use sine. Since we know side b (opposite to angle 2) and hypotenuse c: sin angle 2 = 14/18 angle 2 = sin⁻¹(14/18) = 51.06°

Always check your calculator mode (degrees vs. radians) and remember that sin and cos values are never greater than 1!

More Triangle Applications

Using trigonometry to find missing pieces of triangles is like solving a puzzle. Let's continue with another example.

Example 4: To find angle 1 when a = 1 and b = 3, use tangent. Since we know side a (opposite) and side b (adjacent): tan angle 1 = 1/3 angle 1 = tan⁻¹(1/3) = 18.43°

Important reminders when solving triangle problems:

• Always check that your calculator is in the correct mode (degrees vs. radians)
• If sin or cos values are greater than 1, you've made an error (likely with the hypotenuse)
• Include the degree symbol (°) when referring to angles, or else you're talking about radians

These fundamental triangle relationships are the building blocks for more complex problems in engineering, physics, and construction. Master these basics, and you'll be ready to tackle those challenges!

Circular Functions and Angle Positions

Angles in standard position start on the positive x-axis and rotate counterclockwise. Think of it like a spinner with the center at the origin of a coordinate plane.

Coterminal angles share the same initial and terminal sides but have different measurements. For example, 40° and 400° are coterminal because 400° = 40° + 360°.

The reference angle is the acute angle formed between the terminal side and the x-axis. It helps us find trig values for any angle!

For example, with 312°:

1. Find where it lands (in the 4th quadrant)
2. Calculate the reference angle: 360° - 312° = 48°
3. Draw the angle to visualize it

With negative angles like -153°:

1. Find a coterminal positive angle: -153° + 360° = 207°
2. Locate it (in the 3rd quadrant)
3. Calculate the reference angle: 207° - 180° = 27°

🌟 Knowing the reference angle is like having a cheat code for trig values! Any angle has the same trig value magnitudes as its reference angle (though the signs may differ).

The Unit Circle: Special Triangles

The unit circle is your secret weapon for finding exact trig values! It's built around two special right triangles:

45°-45°-90° Triangle:

• When placed on the unit circle, this forms angles at 45° and 135°
• Both legs have length x
• The hypotenuse is x√2
• At 45°, both x and y coordinates equal √2/2

30°-60°-90° Triangle:

• When placed on the unit circle, this forms angles at 30°, 60°, 150°, etc.
• If the shortest leg is x, the other leg is x√3
• The hypotenuse is 2x
• At 30°, x=√3/2 and y=1/2
• At 60°, x=1/2 and y=√3/2

These special triangles give us exact values for key angles around the unit circle, which means you can find exact trig values without a calculator!

💡 These special triangles are worth memorizing! They'll save you tons of time when working with common angles like 30°, 45°, and 60°.

Unit Circle Coordinates and Trig Values

The unit circle connects geometry and trigonometry in one elegant picture. With a radius of 1, any point (x,y) on the circle corresponds to an angle θ from the positive x-axis.

The key insight: for any angle θ,

• x = cos θ
• y = sin θ

This means when you see an angle like 30° on the unit circle, the coordinates (√3/2, 1/2) tell you that:

• cos 30° = √3/2
• sin 30° = 1/2

The signs of these values depend on the quadrant:

• Quadrant I (0°-90°): Both x and y are positive
• Quadrant II (90°-180°): x is negative, y is positive
• Quadrant III (180°-270°): Both x and y are negative
• Quadrant IV (270°-360°): x is positive, y is negative

Remember that points at 0°, 90°, 180°, and 270° are (1,0), (0,1), (-1,0), and (0,-1) respectively.