# Unit 4: Trigonometry ## Angle Theorems 1. ```Transversal Parallel Line Theorems``` (TPT) a. Alternate Angles are Equal ```(Z-Pattern)``` b. Corresponding Angles Equal ```(F-Pattern)``` c. Interior Angles add up to 180 ```(C-Pattern)``` -

2. ```Supplementary Angle Theorem``` (SAT) - When two angles add up to 180 degrees -

3. ```Opposite Angle Theorem (OAT)``` (OAT) - Two lines intersect, two angles form opposite. They have equal measures -

4. ```Complementary Angle Theorem``` (CAT) - The sum of two angles that add up to 90 degrees -

5. ```Angle Sum of a Triangle Theorem``` (ASTT) - The sum of the three interior angles of any triangle is 180 degrees -

6. ```Exterior Angle Theorem``` (EAT) - The measure of an exterior angle is equal to the sum of the measures of the opposite interior angles -

7. ``` Isosceles Triangle Theorem``` (ITT) - The base angles in any isosceles triangle are equal -

8. ```Sum of The Interior Angle of a Polygon``` - The sum of the interioir angles of any polygon is ```180(n-2)``` or ```180n - 360```, where ```n``` is the number of sides of the polygon -

9. ```Exterior Angles of a Convex Polygon``` - The sum of the exterior angle of any convex polygon is always ```360 degrees``` -

## Congruency `Congruent`: Same size and shape ### Side-Side-Side (SSS) If three sides of a triangle are respectively equal to the three sides of another triangle, then the triangles are congruent

### Side-Angle-Side (SAS) 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.

### Angle-Side-Angle (ASA) 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.

## Similar Triangles `Similar`: Same shape but different sizes (one is an enlargement of the other) ### Properties Lets say we have $`\triangle ABC \sim \triangle DEF`$ 1. Corresponding angles are **equal** - $`\angle A = \angle D`$ - $`\angle B = \angle E`$ - $`\angle C = \angle F`$ 2. Corresponding side are **proportional**. - $`\dfrac{AB}{DE} = \dfrac{AC}{DF} = \dfrac{BC}{EF}`$ 3. Proportional Area - Let $`k`$ be the **scale factor**, when concerning for triangle area, if the triangle area can be defined as $`\dfrac{bh}{2}`$, then by using the smaller triangles side lengths our big triangle's area is equal to $`\dfrac{k^2bh}{2}`$. Similar equations and agruments can be dervied from this ### Side-Side-Side similarity (RRR $`\sim`$) Three pairs of corresponding sides are in the **same ratio**

### Side-Angle-Side similarity (RAR $`\sim`$) Two pairs of corresponding sides are proportional and the **contained** angle are equal.

### Angle-Angle similarity (AA $`\sim`$) 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.

## Primary Trigonometric Ratios |Part Of Triangle|Property| |:---------------|:-------| |Hypotenuse|The longest side of the right triangle. it is across the $`90^o`$ (right angle)| |Opposite|The side opposite to the reference angle| |Adjacent|The side next to the reference agnle| **Remember**: Primary trigonometric ratios are only used to find the **acute** angles or sides of a **right-angled** triangle ### SOH CAH TOA **SINE** $`\sin \theta = \dfrac{\text{Opposite}}{\text{Hypotenuse}}`$ **COSINE** $`\cos \theta = \dfrac{\text{Adajacent}}{\text{Hypotenuse}}`$ **TANGENT** $`\tan \theta = \dfrac{\text{Opposite}}{\text{Adajacent}}`$ ## Angle Of Elevation And Depression | |Angle of Elevation|Angle of Depression| |:---------|:-----------------|:------------------| |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| |Diagram|

| We can see that **Angle of Elevation = Angle of Depression** in the diagram below (Proven using Z-pattern)

## Sine Law In any $`\triangle ABC`$: $`\dfrac{\sin A}{a} = \dfrac{\sin B}{b} = \dfrac{\sin C}{c}`$ or $`\dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C}`$ We can derive the formula further to get: - $`\dfrac{\sin A}{\sin B} = \dfrac{a}{b}`$ - $`\dfrac{\sin A}{\sin C} = \dfrac{a}{c}`$ - $`\dfrac{\sin B}{\sin C} = \dfrac{b}{c}`$ Also, for some trigonometry identities: - $`\tan x = \dfrac{\sin x}{\cos x}`$ - $`\sin^2 A + \cos^2 A = 1`$ **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.** ### Ambiguous Case 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 give 0, 1, or 2 solutions. In the ambiguous case, if $`\angle A, a, b`$ are given, the height of the triangle is $`h= b\sin A`$ |Case|If $`\angle A`$ is **acute**|Condition|# & Type of triangles possible| |:---|:---------------------------|:--------|:-----------------------------| |1 |

|$`a = h`$|one triangle exists| |3 |

|$`a \ge b`$|one triangle exists| |Case|If $`\angle A`$ is **obtuse**|Condition|# & Type of triangles possible| |:---|:----------------------------|:--------|:-----------------------------| |5 |

|$`a \gt b`$|one triangle exists| ## Cosine Law In any $`\triangle ABC`$, $`c^2 = a^2 + b^2 - 2ab\cose C`$ **If you are given 3 sides or 2 sides and the contained angle of an `oblique triangle`, then use the consine law** ## Directions `Bearings`: **Always** start from **North**, and goes **clockwise** `Direction`: Start from the first letter (N, E, S, W), and go that many degrees directly to the second letter (N, E, S, W) **Note:** Northeast, southeast, northwest etc. all have 45 degrees to the left or the right from their starting degree (0, 90, 180, 270) ## 2D Problems **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 ## 3D problems **Note:** Use angle theorems to find bearing/direction angle, and to help with the problem in general. Apply sine law, cosine law, and primary trigonometric ratios whenever necessary.