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Update Unit 4: Trigonometry.md
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# Unit 4: Trigonometry
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## Angle Theorems
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1. ```Transversal Parallel Line Theorems``` (TPT)
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@ -55,12 +57,18 @@
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If three sides of a triangle are respectively equal to the three sides of another triangle, then the triangles are congruent
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<img src="https://media.cheggcdn.com/study/51f/51f6fea6-a0df-4da9-ad5c-f9663291a22f/DC-2411V1.png" width="500">
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### Side-Angle-Side (SAS)
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If two sides and the **contained** angle of a triangle are respectively equal to two sides and the **contained** angle of another triangle, then the triangles are congruent.
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<img src="https://dr282zn36sxxg.cloudfront.net/datastreams/f-d%3A77ca1d4d5a96259729f68cffe461702e9c92b5abb6018335683fa888%2BIMAGE_TINY%2BIMAGE_TINY.1" width="300">
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### Angle-Side-Angle (ASA)
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If two angles and the **contained** side of a triangle are respectively equal to two angles and the **contained** side of another triangle, then the triangles are congruent.
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<img src="https://www.onlinemath4all.com/images/trianglecongruenceandsimilarity4.png" width="500">
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## Similary Triangles
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`Similar`: Same shape but different sizes (one is an enlargement of the other)
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@ -83,10 +91,101 @@ our big triangle's area is equal to $`\dfrac{k^2bh}{2}`$. Similar equations and
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Three pairs of corresponding sides are in the **same ratio**
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<img src="https://dr282zn36sxxg.cloudfront.net/datastreams/f-d%3Ac6e3b786fdc6105d64b086efcfa48c529b91cbb087f6ba3bc60b9f9f%2BIMAGE_TINY%2BIMAGE_TINY.1" width="500">
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### Side Angle Side similarity (RAR $`\sim`$)
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Two pairs of corresponding sides are proportional and the **contained** angle are equal.
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<img src="https://dr282zn36sxxg.cloudfront.net/datastreams/f-d%3A3c5ec26759a40dc1d524b1c5af8864d5e87135063ce6f4e75d37af4d%2BIMAGE_TINY%2BIMAGE_TINY.1" width="500">
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### Angle-Angle similarity (AA $`\sim`$)
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Two pairs of corresponding angles are equal.
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Two pairs of corresponding angles are equal. In the diagram below, we can solve for the missing angle using Angle Sum Of A Triangle Theorem (ASTT) and see that those 2 triangle's angles are equal.
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<img src="https://www.onlinemath4all.com/images/angleanglesimilarity2.png" width="500">
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## Primary Trigonometry Ratios
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|Part Of Triangle|Property|
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|:---------------|:-------|
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|Hypotenuse|The longest side of the right triangle. it is across the $`90^o`$ (right angle)|
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|Opposite|The side opposite to the reference angle|
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|Adjacent|The side next to the reference agnle|
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**Remember**: Primary Trigonometry ratios are only used to find the **acute** angles or sides of a **right-angled** triangle
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### SOH CAH TOA
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**SINE** $`\sin \theta = \dfrac{\text{Opposite}}{\text{Hypotenuse}}`$
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**COSINE** $`\cos \theta = \dfrac{\text{Adajacent}}{\text{Hypotenuse}}`$
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**TANGENT** $`\tan \theta = \dfrac{\text{Opposite}}{\text{Adajacent}}`$
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## Angle Of Elevation And Depression
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| |Angle of Elevation|Angle of Depression|
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|:---------|:-----------------|:------------------|
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|Definition|**Angle of Elevation** is the angle from the horizontal looking **up** to some object|**Angle of Depression** is the angle frorm the horizontal looking **down** to some object|
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|Diagram|<img src="https://www.varsitytutors.com/assets/vt-hotmath-legacy/hotmath_help/topics/angle-of-elevation/angle-of-elevation-image001.gif" width="300">|<img src="https://www.varsitytutors.com/assets/vt-hotmath-legacy/hotmath_help/topics/angle-of-elevation/angle-of-elevation-image002.gif" width="300">|
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We can see that **Angle of Elevation = Angle of Depression** in the diagram below (Proven using Z-pattern)
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<img src="https://www.varsitytutors.com/assets/vt-hotmath-legacy/hotmath_help/topics/angle-of-elevation/angle-of-elevation-image003.gif" width="300">
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## Sine Law
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In any $`\triangle ABC`$: $`\dfrac{\sin A}{a} = \dfrac{\sin B}{b} = \dfrac{\sin C}{c}`$ or $`\dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C}`$
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We can derive the formula further to get:
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- $`\dfrac{\sin A}{\sin B} = \dfrac{a}{b}`$
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- $`\dfrac{\sin A}{\sin C} = \dfrac{a}{c}`$
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- $`\dfrac{\sin B}{\sin C} = \dfrac{b}{c}`$
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Also, for some trigonometry identities:
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- $`\tan x = \dfrac{\sin x}{\cos x}`$
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- $`\sin^2 A + \cos^2 A = 1`$
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**If you are finding the sides or agnles of an `oblique triangle` given 1 side, its opposite angle and one other side or angle, use the sine law.**
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### Ambiguous Case
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The ambiguous case arises in the SSA or (ASS) case of an triangle, when you are given angle side side. The sine law calculation may need to 0, 1, or 2 solutions.
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In the ambigouous case, if $`\angle A, a, b`$ are given, the height of the triangle is $`h= b\sin A`$
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|Case|If $`\angle A`$ is **acute**|Condition|# & Type of triangles possible|
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|:---|:---------------------------|:--------|:-----------------------------|
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|1 |<img src="https://encrypted-tbn0.gstatic.com/images?q=tbn:ANd9GcSQfCqnJUFFQ9UQHoYT-CXd614RTXNGL_R5QNomKDCaDV3Nja4g&s" width="200">|$`a \lt h`$|no triangle exists|
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|2 |<img src="https://www.analyzemath.com/Triangle/sine_law_2.gif" width="300">|$`a = h`$|one triangle exists|
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|3 |<img src="http://www.technologyuk.net/mathematics/trigonometry/images/trigonometry_0078.gif" width="300">|$`h \lt a \lt b`$|two triangle exist (one acute triangle, one obtuse triangle)|
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|4 |<img src="http://jwilson.coe.uga.edu/EMT668/EMAT6680.2001/Mealor/EMAT%206700/law%20of%20sines/Law%20of%20Sines%20ambiguous%20case/image3.gif" width="300">|$`a \ge b`$|one triangle exists|
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|Case|If $`\angle A`$ is **obtuse**|Condition|# & Type of triangles possible|
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|:---|:----------------------------|:--------|:-----------------------------|
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|5 |<img src="http://jwilson.coe.uga.edu/EMT668/EMAT6680.2001/Mealor/EMAT%206700/law%20of%20sines/Law%20of%20Sines%20ambiguous%20case/image5.gif" width="200">|$`a \le b`$|no triangle exists|
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|6 |<img src="http://www.gradeamathhelp.com/image-files/ambiguous-case-side-length.gif" width="200">|$`a \gt b`$|one triangle exists|
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## Cosine Law
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In any $`\triangle ABC`$, $`c^2 = a^2 + b^2 - 2ab\cose C`$
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**If you are given 3 sides or 2 sides and the contained angle of an `oblique triangle`, then use the consine law**
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## Directions
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`Bearins`: **Always** start from **North**, and goes **clockwise**
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`Direction`: Start from the first letter (N, E, S, W), and go that many degrees to the second letter (N, E, S, W)
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**Note:** Northeast, Southeast, NorthWest etc. all have 45 degrees to the left or the right from their starting degree (0, 90, 180, 270)
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## 2D Problems
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**Note:** Watch out for the case where you don't know which side the 2 things (buildings, boats etc) are, they can result in 2 answers
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## 3D problems
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**Note:** Use angle theorems to find bearing/direction angle, and to help with the problem in general. Apply sine law, cosine law, and primary trigonometry ratios whenever necessary.
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