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### 2002-E: Trigonometry 2: Trigonometric Functions

• Topic Cluster: Pure Mathematics
• Topic: Geometry
• Objective: Solve for unknown sides of a triangle using sine, cosine, and tangent, including problems in which the unknown side is in the denominator
• Content:
• Level: 2

We typically label angles by the Greek letter $$\theta$$, pronounced "theta." Once we have singled out a particular angle, each side has a new name:

• A is the opposite of $$\theta$$.
• B is the adjacent of $$\theta$$.
• C is the hypotenuse.
• $$\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$$
• $$\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}$$
• $$\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$$

Whenever we know what an angle is, we can use the three trigonometric functions: the sine, the cosine, and the tangent. To remember these three formulas, we use the silly word "SOHCAHTOA," which stands for: "Sine is opposite over hypotenuse; cosine is adjacent over hypotenuse; tangent is opposite over adjacent."

#### Problems

In each case, use one of the trigonometric functions to solve for the unknown side. For full credit, you must write the formula before using it.

#### Variable in Denominator

In some trigonometry problems, the unknown variable will be located in the denominator! In this case, the most straightforward method is to first multiply both sides of the equation by the unknown variable, then solve for it:

##### Example:

$$\require{enclose} 48 = \frac{76}{X} \\ 48 \cdot X = \frac{76}{\enclose{horizontalstrike}{X}} \cdot \enclose{horizontalstrike}{X} \\ 48 \cdot X = 76 \\ X = \frac{76}{48} = 1.583$$

Solve for $$x$$ in each of the problems below. Please give all answers to 4 significant figures of accuracy.

1. $$49 = \frac{65}{x}$$
2. $$66 = \frac{82}{x}$$
3. $$42 = \frac{25}{x}$$

#### Saying the Names Right!

• Whenever we use the sine function, we write the abbreviation $$\sin$$, but when reading this out loud you always say "sine!"
• Whenever we use the cosine function, we write the abbreviation $$\cos$$, but when reading this out loud you always say "cosine!"
• Whenever we use the tangent function, we write the abbreviation $$\tan$$, but when reading this out loud you always say "tangent!"
1. How do you read out loud the following equation: $$\sin 90^{\circ} = 1$$.
2. How do you read out loud the following equation: $$\cos 90^{\circ} = 0$$.
3. How do you read out loud the following equation: $$\tan 45^{\circ} = 1$$.

#### Extra Practice

(Not required to receive full credit for this assignment.)