3.4 KiB
Unit 3
Review
Some special ratios you should know:
\(`\csc \theta = \dfrac{1}{sin \theta}`\)
\(`\sec \theta = \dfrac{1}{\cos \theta}`\)
\(`\cot \theta = \dfrac{1}{\tan \theta}`\)
Special Angles
These are the angles in trigonometery that have “nice” solutions, and does not require a calculator.
| Degrees | \(`0^o`\) | \(`30^o`\) | \(`45^o`\) | \(`60^o`\) | \(`90^o`\) |
|---|---|---|---|---|---|
| \(`\sin \theta`\) | \(`0`\) | \(`\dfrac{1}{2}`\) | \(`\dfrac{\sqrt{2}}{2}`\) | \(`\dfrac{\sqrt{3}}{2}`\) | \(`1`\) |
| \(`\cos \theta`\) | \(`1`\) | \(`\dfrac{\sqrt{3}}{2}`\) | \(`\dfrac{\sqrt{2}}{2}`\) | \(`\dfrac{1}{2}`\) | \(`0`\) |
| \(`\tan \theta`\) | \(`0`\) | \(`\dfrac{1}{sqrt{3}}`\) | \(`1`\) | \(`\sqrt{3}`\) | UNDEFINED |
Standard Position and Co-terminal Angles
Standard position consists of 2 arms and an angle. The
arm that is ALWAYS on the x-axis is called the
initial arm. And the other arm is called the
terminal arm.
Positive angles go counter-clockwise direction (to the right), and negative angles go clockwise direction (to the left).
Co-terminal Angles are angles whose terminal arms have
the same standard position. Any 2 angles that are \(`360^o`\) apart are considered
Co-terminal angles.
The CAST Rule
Principal Angle \(`\theta`\): This is the angle usually given
in the question. It is the counter-clockwise angle bewteen the initial
arm and the terminal arm of an angle in standard position.
Related Acute Angle or Reference Angle (\(`\alpha`\)): The angle between the terminal
arm and the x-axis. Note that this angle is always in
the range \(`0^o \le \alpha \le
90^o`\).
The CAST simply determines the positive/negatie signs of the result of a trig function of the related angle.
Simple evaluate the related angle with the respective trigonmetery function, and add a negative sign according to the picture above.
Solving Trigonmetric Equations
Just a few simple steps.
- Simplifiy the expression to make all the trig functions on one side, and the constants on the other. Makes sure not to divide or omit trig functiosn involving the variable as you might be omitting solutions.
- Factor and simplify the expression, and state ALL
possible solutions using
Co-terminal angles. - Profit!
Degrees and Radians
A few formulas:
To convert degrees to radians, multiply the degrees by \(`\dfrac{\pi}{180^o}`\).
To convert radians to degrees, multiply the radians by \(`\dfrac{180^o}{\pi}`\)
To find the arc length (\(`s`\)) of the circle described by the angle and radius (or commonly known as subtended by the angle measure): \(`s = r\theta`\), where \(`\theta`\) is described in radians.
In the absence of the degree symbol, the angle must be assumed to be in radians. It is also useful to know that \(`\pi = 180^o`\) and \(`2\pi = 360^o`\)
To find the RAA in terms of radians, follow the table below.
| Quadrant | Quadrant 1 (All) | Quadrant 2 (Sin) | Quadrant 3 (tan) | Quadrant 4(cos) |
|---|---|---|---|---|
| Step to do to get \(`\alpha`\) | \(`\alpha = \theta`\) | \(`\alpha = \pi - \theta`\) | \(`\alpha = \theta - \pi`\) | \(`\alpha = 2\pi - \theta`\) |
To find the area of a sector of a circle, the formula is as follows.
\(`\text{Sector Area } = (\text{Area of Circle})(\% \text{ Of Circle shaded in})`\)
\(`A = (\pi r^2)(\dfrac{\theta}{2\pi})`\)
\(`A = \dfrac{1}{2}\theta r^2`\)