# Law of Sine and Cosine Worksheets

Until now, we have been working primarily with right triangles and finding missing measures within them (side lengths and angles) using trigonometric functions. You will find yourself in many situations where there are no right triangles to use as a reference point to apply to the remainder of the environment. The law of Sine and Cosine allow us to fid missing measure in other types of triangles such as equilaterals, isosceles, and scalene. A series of printable worksheets that help students learn to use these functions as tools to help reveal information about right triangles.

### Aligned Standard: High School Geometry - HSG-SRT.D.10

- Missing Sides Step-by-step Lesson -We get familiar with our favorite triangle theorems, once again.
- Guided Lesson - I tried to touch on all the major applications of these problems that appear on standardized tests. See how I did.
- Guided Lesson Explanation - I used a slight smaller font here to reduce the number of pages that you need to print.
- Practice Worksheet - A series of solid pages for you to work on here.
- Matching Worksheet - Match the parts of the theorems to help this one make more sense and finish the problems faster.

- Answer Keys - These are for all the unlocked materials above.

### Homework Sheets

We hit every part of a triangle with this one.

- Homework 1 - The law of sin is based on the proportionality of side and angle in triangle. The law states that for the angle of a non right angle, each angle of the triangle has the same ratio of angle measure to sine value.
- Homework 2 - We have to write down the givens. Use the law of cosines, substitute, and simplify. Isolate cos A find the inverse.
- Homework 3 - We have to write down what is known. Alternative forms of law of cosines can be substituted.

### Practice Worksheets

We give you 2 angles and 1 side of a triangle, you tell us the rest.

- Practice 1 - Use the Law of Sine and the Law of Cosine to find the missing sides and angles of each triangle. BC = 14, A = 30°, and B = 90°
- Practice 2 - Find the value of ∠ P, ∠ L, and ∠ A.
- Practice 3 - Substitute the known value into the law of Sine to find the missing lengths.

### Math Skill Quizzes

Find all the missing sides and measures.

- Quiz 1 - The law of sine is based on the proportionality of the side and angle in a triangle.
- Quiz 2 - Find that missing value.
- Quiz 3 - Figure out which function best fits your need here.

### How To Use the Law of Sine and Cosine

Trigonometry is the branch of mathematics that deals with the measurements and the relationships between the side lengths and the angles of a triangle. When we are dealing with right-angled triangles, we use the following trigonometric functions for our calculations; to locate missing side lengths and angles of measure. The commonly used functions are: sin Β = Opposite/Hypotenuse, cos Β = Adjacent/Hypotenuse, tan Β = Opposite/Adjacent. And their inverses, given by; csc Β =(Hypotenuse )/Opposite, sec Β =Hypotenuse/Adjacent, cot Β =Adjacent/Opposite.

However, when we are dealing with other types of triangles, such as isosceles, scalene, and equilateral, these six trigonometric functions cannot be applied. This is where the sine and the cosine rule enter trigonometry and can help us out tremendously because we will be able to find missing measures on those triangles using these rules.

**Law of Sines** - The law of sines is given by; a/sin A =b/sin B c/sin C.

This basically tells us that when you divide a side by the sine of angle A it is equivalent to quotient of side b and the sine of angle B, as well as the quotient of side C and the sine of angle C. This is very helpful for finding unknown angle measurements in these types of triangles. There are two conditions when you can use the sine rule; When two angles and one side is given. When two sides and one non-included angle is given, it is sometimes called the ambiguous case.

** Law of Cosine** - This is often referred to as the Cosine Rule. This will help us determine the measure of lengths of sides and angle measurements in many non-right-angled triangles. The cosine rule is given by;

c^{2} = a^{2} + b^{2} - 2ab cos(C)

The lowercase letters (a, b, and c) all represent sides of the triangle. The UPPERCASE C represents the angle that is opposite side C. There are two conditions when you can use the cosine rule, when two sides and one included angle is given. All three sides are given.