Algebra 2 Graphs of Trigonometric Functions and Identities Lesson (Free PDF)

Algebra 2 math lesson: Graphs of Trigonometric Functions and Identities – Maths PDF

Cover of the book Algebra 2 Math Lessons (PDF)

Download as a PDFThe Algebra 2 math lessons bookPrintable PDFSee the book ›

Sommaire

This algebra 2 trig graphs lesson for Algebra 2 covers graphs of trigonometric functions and identities (Common Core F-TF.B.5, F-TF.C.8, F-IF.C.7e), with a free PDF to print.

Sine and cosine graphs are periodic waves with period 2π. For y = A sin(B(x – C)) + D, the amplitude is |A|, the period is 2π/B, the shift is C, and the midline is y = D. The identity sin² θ + cos² θ = 1 links sine and cosine.

Algebra 2 · Common Core State Standards: F-TF.B.5, F-TF.C.8, F-IF.C.7e

Key facts in this algebra 2 trig graphs lesson

Key facts

Sine and cosine have period 2π and range -1 to 1.
y = A sin(B(x – C)) + D: amplitude |A|, period 2π/B, shift C, midline D.
Model cycles with a sinusoid: midline = average of max and min.
sin² θ + cos² θ = 1.
The quadrant gives the sign of the missing trig value.

What do sine and cosine graphs look like?

The graphs of \(y = \sin x\) and \(y = \cos x\) are smooth waves that repeat every \(2\pi\). They are periodic: \(f(x + 2\pi) = f(x)\). Sine starts at 0 and rises, while cosine starts at its maximum 1. Cosine is sine shifted left by \(\frac{\pi}{2}\).

y = sin x (blue) and y = cos x (orange) both have period 2π and amplitude 1.

Sinusoidal functions

A sinusoidal function has the form \(y = A\sin(B(x – C)) + D\) or \(y = A\cos(B(x – C)) + D\). The amplitude is \(|A|\), the period is \(\frac{2\pi}{B}\), the phase shift is \(C\) (right if positive), and the midline is \(y = D\).

y = 2 sin x + 1: midline y = 1, amplitude 2, period 2π.

For \(y = 2\sin x + 1\), the midline is \(y = 1\), the maximum is \(1 + 2 = 3\), the minimum is \(1 – 2 = -1\), and the period is \(2\pi\). A negative \(A\) reflects the wave over the midline.

How do you model periodic situations?

Many real-world cycles are sinusoidal: tides, daylight hours, temperature, and a rider on a Ferris wheel. Match each feature: half the distance between the maximum and minimum is the amplitude, their average is the midline, and the time for one cycle is the period, from which \(B = \frac{2\pi}{\text{period}}\). A wheel with radius 20 feet, center 25 feet above the ground, and a 30 second turn has height \(h(t) = 25 – 20\cos\left(\frac{\pi}{15}t\right)\).

Reading the parameters of a sinusoidal function.

What is the Pythagorean identity?

A point on the unit circle at angle \(\theta\) is \((\cos\theta, \sin\theta)\). By the Pythagorean theorem, \(\sin^{2}\theta + \cos^{2}\theta = 1\) for every angle. If you know one of sine or cosine and the quadrant, you can find the others. If \(\sin\theta = \frac{3}{5}\) and \(\theta\) is in quadrant II, then \(\cos^{2}\theta = 1 – \frac{9}{25} = \frac{16}{25}\). Cosine is negative in quadrant II, so \(\cos\theta = -\frac{4}{5}\) and \(\tan\theta = -\frac{3}{4}\).

The right triangle in the unit circle: cos² θ + sin² θ = 1.

Algebra 2 trig graphs lesson: questions and answers

What is the amplitude of a sine function?

The amplitude is half the distance between the maximum and the minimum. For y = A sin x it equals the absolute value of A.

How do you find the period of y = sin(Bx)?

The period is 2π divided by B. For y = sin(3x) the period is 2π/3, so the graph completes three cycles from 0 to 2π.

What does the Pythagorean identity say?

It says sine squared of an angle plus cosine squared of the same angle equals 1. It follows from the unit circle having radius 1.

How do you find cosine if you know sine and the quadrant?

Use cos² θ = 1 – sin² θ, take the square root, and choose the sign that matches the quadrant.

What is the midline of a trig graph?

The midline is the horizontal line halfway between the maximum and minimum, y = D in y = A sin(B(x – C)) + D.

How to use this algebra 2 trig graphs lesson

Read the definitions first, then follow the worked method line by line and redo the example on paper without looking. When every step makes sense, move on to the worksheet, then check yourself with the quiz.

Download the algebra 2 trig graphs lesson as a PDF with the button below and keep it with your notes.

All Algebra 2 chapters: Algebra 2 math.

Voter.. post

Free printable PDF