Trigonometry Radar Review Worksheet
TRIGONOMETRY RADAR
Tactical Data Sheet (Reference)
NAVIGATOR:
TIMESTAMP:
I. COORDINATE VECTORS
Standard Position: Vertex at (0,0), initial side on pos. x-axis.
\(r = \sqrt{x^2 + y^2}\)
\(x^2 + y^2 = r^2\)
\(\sin = y/r\)
\(\cos = x/r\)
\(\tan = y/x\)
\(\csc = r/y\)
\(\sec = r/x\)
\(\cot = x/y\)
II. SIGNAL MODELING
\(y = a\sin(b(x-c)) + d\)
- Amplitude: \(|a| = \frac{max - min}{2}\)
- Period: \(P = \frac{2\pi}{|b|}\) or \(P = \frac{\pi}{|b|}\) (tan)
- Midline: \(y = d = \frac{max + min}{2}\)
- Phase Shift: \(c\) (Right if \(-\), Left if \(+\))
III. CORE IDENTITIES
Pythagorean:
\(\sin^2\theta + \cos^2\theta = 1\)
Double Angle / Power Reduc:
\(\sin(2\theta) = 2\sin\theta\cos\theta\)
\(\cos^2\theta = \frac{1+\cos(2\theta)}{2}\)
\(\sin^2\theta = \frac{1-\cos(2\theta)}{2}\)
IV. RANGE RESTRICTIONS
| Function | Range |
|---|
| arcsin(x) | \( [-\pi/2, \pi/2] \) |
| arccos(x) | \( [0, \pi] \) |
| arctan(x) | \( (-\pi/2, \pi/2) \) |
V. POLAR TRACKING PROTOCOLS
Conversion Equations:
\(x = r \cos \theta\)
\(y = r \sin \theta\)
\(r^2 = x^2 + y^2\)
\(\tan \theta = y/x\)
Curve Patterns:
- Rose Curves: \(r = a\sin(n\theta)\) or \(r = a\cos(n\theta)\)
- If \(n\) is odd: \(n\) petals. If \(n\) is even: \(2n\) petals.
- Circles: \(r = a\cos\theta\) (Center on x-axis) or \(r = a\sin\theta\) (Center on y-axis)
REF DATA SHEET 3.20 Check for internal alignment before proceeding
MISSION TASKS: SECTION 1
UNIT 3 REVIEW
Select the best answer for each coordinate check. Show your logic in the tactical grid areas provided. Standard calculators are permitted.
1
An angle \(\theta\) is in standard position. The terminal side of \(\theta\) passes through the point \((-3, 4)\). What is the value of \(\sec \theta\)?
A \(-\frac{5}{3}\)
B \(-\frac{3}{5}\)
C \(\frac{4}{5}\)
D \(\frac{5}{4}\)
2
What is the period of the function \(f(x) = 3\sin\left(\frac{\pi}{4}x - 2\right) + 5\)?
A \(\frac{\pi}{4}\)
B \(4\)
C \(8\)
D \(2\pi\)
3
If \(\cos \theta = \frac{1}{2}\) and \(\frac{3\pi}{2} < \theta < 2\pi\), what is the value of \(\sin \theta\)?
A \(\frac{\sqrt{3}}{2}\)
B \(-\frac{\sqrt{3}}{2}\)
C \(\frac{\sqrt{2}}{2}\)
D \(-\frac{1}{2}\)
4
Which of the following is equivalent to \(\frac{\sin^2 \theta}{\cos \theta} + \cos \theta\)?
A \(\sin \theta\)
B \(\sec \theta\)
C \(\csc \theta\)
D \(1\)
5
A ferris wheel has a radius of 25 feet and its center is 30 feet above the ground. If the wheel completes one revolution every 40 seconds, which function models the height \(h(t)\) of a rider in feet \(t\) seconds after they are at the lowest point?
A \(h(t) = 25\cos\left(\frac{\pi}{20}t\right) + 30\)
B \(h(t) = -25\cos\left(\frac{\pi}{20}t\right) + 30\)
C \(h(t) = 25\sin\left(\frac{\pi}{20}t\right) + 30\)
D \(h(t) = -25\sin\left(\frac{\pi}{20}t\right) + 30\)
6
Find the domain of the function \(f(x) = \tan(2x)\).
A All real numbers \(x\) such that \(x \neq \frac{\pi}{2} + k\pi\) for any integer \(k\).
B All real numbers \(x\) such that \(x \neq \frac{\pi}{4} + \frac{k\pi}{2}\) for any integer \(k\).
C All real numbers \(x\) such that \(x \neq k\pi\) for any integer \(k\).
D All real numbers \(x\) such that \(x \neq \frac{k\pi}{2}\) for any integer \(k\).
7
What is the value of \(\arcsin\left(\sin\frac{5\pi}{4}\right)\)?
A \(\frac{5\pi}{4}\)
B \(\frac{\pi}{4}\)
C \(-\frac{\pi}{4}\)
D \(\frac{3\pi}{4}\)
8
Convert the polar point \((4, \frac{5\pi}{6})\) to rectangular coordinates \((x, y)\).
A \((-2\sqrt{3}, 2)\)
B \((2, -2\sqrt{3})\)
C \((-2, 2\sqrt{3})\)
D \((2\sqrt{3}, -2)\)
9
Which polar equation represents a circle passing through the origin with its center on the positive y-axis?
A \(r = 4\cos\theta\)
B \(r = 4\sin\theta\)
C \(r = 4\)
D \(r = \theta\)
10
A rose curve is given by \(r = 3\sin(2\theta)\). How many petals does this curve have?
A 2
B 3
C 4
D 8
11
Simplify the expression \(\frac{1 - \cos^2 x}{\sin x}\).
A \(\cos x\)
B \(\sin x\)
C \(\tan x\)
D \(1\)
12
The point \((x, y) = (-1, -1)\) in rectangular coordinates can be represented in polar coordinates \((r, \theta)\) as:
A \((\sqrt{2}, \frac{\pi}{4})\)
B \((\sqrt{2}, \frac{5\pi}{4})\)
C \((2, \frac{3\pi}{4})\)
D \((2, \frac{7\pi}{4})\)
13
Which function has an amplitude of 4 and a phase shift of \(\frac{\pi}{2}\) to the right?
A \(f(x) = 4\sin(x + \frac{\pi}{2})\)
B \(f(x) = 4\sin(x - \frac{\pi}{2})\)
C \(f(x) = \frac{1}{4}\sin(x - \frac{\pi}{2})\)
D \(f(x) = \frac{1}{4}\sin(x + \frac{\pi}{2})\)
14
What is the range of the function \(f(x) = \arctan(x)\)?
A \([-\frac{\pi}{2}, \frac{\pi}{2}]\)
B \((-\frac{\pi}{2}, \frac{\pi}{2})\)
C \([0, \pi]\)
D \((0, \pi)\)
15
Which identity is correctly stated?
A \(\cos(2\theta) = 2\cos^2\theta + 1\)
B \(\sin(2\theta) = \sin\theta\cos\theta\)
C \(\cos^2\theta = \frac{1 + \cos(2\theta)}{2}\)
D \(\sin^2\theta = 1 + \cos^2\theta\)
16
The expression \(\cos(x)\cos(y) - \sin(x)\sin(y)\) is equivalent to:
A \(\cos(x + y)\)
B \(\cos(x - y)\)
C \(\sin(x + y)\)
D \(\sin(x - y)\)
17
What is the maximum value of the function \(r = 3 + 2\cos\theta\)?
A 1
B 3
C 5
D 2
18
In the polar coordinate system, which point is equivalent to \((r, \theta) = (2, \frac{\pi}{3})\)?
A \((-2, \frac{4\pi}{3})\)
B \((2, -\frac{\pi}{3})\)
C \((-2, \frac{\pi}{3})\)
D \((2, \frac{2\pi}{3})\)
19
For the function \(f(x) = \sec x\), at which values of \(x\) do vertical asymptotes occur on the interval \([0, 2\pi]\)?
A \(x = 0, \pi, 2\pi\)
B \(x = \frac{\pi}{2}, \frac{3\pi}{2}\)
C \(x = \frac{\pi}{4}, \frac{5\pi}{4}\)
D No asymptotes exist.
20
Given \(r = f(\theta)\), the conversion from polar to rectangular coordinates follows which rule for the y-coordinate?
A \(y = r \cos\theta\)
B \(y = r \sin\theta\)
C \(y = r \tan\theta\)
D \(y = r^2\)
Signal Locked: End of Transmission
AP Precalc Unit 3 Review | Radar Tech Material 3.20
Trigonometry Radar Answer Key
RADAR DECODER
Unit 3 Review Answer Key
Teacher Resource
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DETAILED TACTICAL BREAKDOWN
Q1: Definition of \(\sec \theta = \frac{r}{x}\). With \((-3, 4)\), \(r=5\) and \(x=-3\). Result: \(-\frac{5}{3}\).
Q2: Period \(P = \frac{2\pi}{|B|}\). Here \(B = \frac{\pi}{4}\), so \(P = 2\pi / (\pi/4) = 8\).
Q5: Rider starts at lowest point (minimum). Midline 30, Amp 25. Min is 5. Function \(h(t) = -25\cos(Bt) + 30\). Period 40 gives \(B = \frac{\pi}{20}\).
Q7: Range restriction for \(\arcsin\) is \([-\frac{\pi}{2}, \frac{\pi}{2}]\). While \(\sin(\frac{5\pi}{4}) = -\frac{\sqrt{2}}{2}\), the principal value is \(-\frac{\pi}{4}\).
Q10: Polar roses \(r = a\sin(n\theta)\). Since \(n=2\) (even), there are \(2n = 4\) petals.
Q15: Power reduction identities are derived from double angle formulas. \(\cos(2\theta) = 2\cos^2\theta - 1 \Rightarrow \cos^2\theta = \frac{1+\cos(2\theta)}{2}\).
Q18: Rotating \(180^\circ\) (\(\pi\)) and reflecting \(r\) yields equivalent points. \((2, \frac{\pi}{3})\) is equivalent to \((-2, \frac{\pi}{3} + \pi) = (-2, \frac{4\pi}{3})\).
AP Precalc Unit 3 Review | Answer Key 3.20 | Confidential Instructor Document
