Precalculus Term 1
Materials for my 1st term Precalculus class at Stuyvesant High School. Textbook: Cohen, [i]Precalculus with Unit Circle Trigonometry[/i], 4th Edition.
Table of Contents
- Trigonometry
- Unit Circle Trigonometry
- Unit Circle: Complementary Angles
- Unit Circle: Sine and Cosine Functions
- Graphing Sine/Cosine with Translations
- Tangent for Acute Angles
- Graphing the Tangent Function
- Lighthouse Function p.553 #69
- Reciprocal Functions
- Inverses of Trigonometric Functions
- Cosine of the Difference of Two Angles
- The Ambiguous Case of SSA
- Vectors
- Projection of a Vector
- Linear Combination of Vectors
- Dot Product of Two Vectors
- Dot Product: Component Form
- Optimization
- Classwork 46 #3
- Polynomials
- Polynomial Inequality
- Rational Functions
- p.308 Example 4
- p.309 Example 6
- p.312 Example 8
- Rational Function with a Hole
Unit Circle Trigonometry
Drag the slider for [math]\theta[/math] to adjust the angle. Notice the coordinates of point [math]P[/math] on the unit circle.
Unit Circle Definition of Trig Functions
We can extend the definitions of sine, cosine and tangent (and their reciprocals) to angles of [i]any[/i] measure, by using the unit circle.[br][br]On the unit circle, if the point [math]P[/math] has coordinates [math]\left(x,y\right)[/math], then from the right triangle we can see that, in Quadrant I:[br][math]x=\cos\theta[/math][br][math]y=\sin\theta[/math][br][math]\frac{y}{x}=\tan\theta[/math][br][br]Therefore, the point where the terminal side of [math]\theta[/math] intersects the the unit circle will have coordinates [math]P\left(\cos\theta,\sin\theta\right)[/math], assuming the angle is in standard position.[br][br]Since the signs of the coordinates depend on the quadrant, the signs of sine, cosine, and tangent will also depend on the quadrant.[br][br]We can determine the coordinates of the point on the unit circle by using [math]\theta'[/math], the [i]reference angle[/i] of [math]\theta[/math]. This is the acute angle formed between the terminal side of [math]\theta[/math] and the [math]x[/math]-axis. Completing the right triangle will get us the ratios for [math]\theta'[/math], which will be identical to the ratios for [math]\theta[/math], except that we may need to attach a negative sign depending on the quadrant.[br][br]You can drag the slider for the angle and see how the red reference angle changes and where the right triangle is that will allow us to find the coordinates of the point on the unit circle and, in turn, the ratios of [math]\theta[/math].
Projection of a Vector
Classwork 46 #3
Polynomial Inequality
As you drag the slider above, watch how the sign of each of the factors of [math]f\left(x\right)[/math] changes. Each factor changes signs exactly once, as you pass its critical value. This is why the roots are called critical values, because they are the [math]x[/math]-values at which the sign of the corresponding factors changes.[br][br]If you want to see what the numerical value of each factor is, then check the "Numbers" box.
p.308 Example 4
Example 4 from p.309[br]Done according to the directions I gave you.