From b0a926384af86ef27211c7073a34b818c816d1cf Mon Sep 17 00:00:00 2001 From: James Su Date: Fri, 17 Apr 2020 00:51:59 +0000 Subject: [PATCH] Update Unit 3: Trigonometric Equations.md --- .../MCR3U7/Unit 3: Trigonometric Equations.md | 67 ++++++++++++++++++- 1 file changed, 66 insertions(+), 1 deletion(-) diff --git a/Grade 10/Math/MCR3U7/Unit 3: Trigonometric Equations.md b/Grade 10/Math/MCR3U7/Unit 3: Trigonometric Equations.md index e90749e..2a23266 100644 --- a/Grade 10/Math/MCR3U7/Unit 3: Trigonometric Equations.md +++ b/Grade 10/Math/MCR3U7/Unit 3: Trigonometric Equations.md @@ -6,4 +6,69 @@ Some special ratios you should know: $`\csc \theta = \dfrac{1}{sin \theta}`$ -$`\sec \theta = \dfrac{1}{\cos \theta}`$ \ No newline at end of file +$`\sec \theta = \dfrac{1}{\cos \theta}`$ + +$`\cot \theta = \dfrac{1}{\tan \theta}`$ + +## 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. + +1. 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. +2. Factor and simplify the expression, and state **ALL** possible solutions using `Co-terminal angles`. +3. 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. + +$`\text{Sector Area } = (\text{Area of Circle})(\% \text{ Of Circle shaded in})`$ + +$`A = (\pir^2)(\dfrac{\theta}{2\pi})`$ +$`A = \dfrac{1}{2}\theta r^2`$