# 40 Unit 8 Homework 6 Trigonometry Review

## Unit 8 Homework 6 Trigonometry Review

### Introduction

Trigonometry is a branch of mathematics that deals with the relationships between the angles and sides of triangles. It is a fundamental concept in many fields, including physics, engineering, and architecture. In this article, we will review the key concepts and formulas of trigonometry to help you prepare for your Unit 8 Homework 6 assignment.

### 1. Trigonometric Ratios

Trigonometric ratios, also known as trigonometric functions, are the ratios of the sides of a right triangle. The three primary trigonometric ratios are sine (sin), cosine (cos), and tangent (tan). These ratios are defined as follows:

- Sine (sin) = Opposite side / Hypotenuse
- Cosine (cos) = Adjacent side / Hypotenuse
- Tangent (tan) = Opposite side / Adjacent side

These ratios are useful in solving various trigonometric problems, such as finding missing angles or sides of a triangle.

### 2. Pythagorean Identity

The Pythagorean identity is a fundamental trigonometric identity that relates the sine and cosine functions. It is derived from the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. The Pythagorean identity is expressed as:

Sin^2(theta) + Cos^2(theta) = 1

This identity is useful in simplifying trigonometric expressions and equations.

### 3. Special Right Triangles

Special right triangles are triangles with angles that have specific values, making their trigonometric ratios easier to calculate. The two most common special right triangles are the 45-45-90 triangle and the 30-60-90 triangle.

- In a 45-45-90 triangle, the angles are 45 degrees, 45 degrees, and 90 degrees. The sides of this triangle are in the ratio 1:1:√2.
- In a 30-60-90 triangle, the angles are 30 degrees, 60 degrees, and 90 degrees. The sides of this triangle are in the ratio 1:√3:2.

Knowing the ratios of the sides in these special triangles can help simplify trigonometric calculations.

### 4. Trigonometric Identities

Trigonometric identities are equations that are true for all values of the variables involved. They can be used to simplify trigonometric expressions and solve trigonometric equations. Some of the most commonly used trigonometric identities include:

- Reciprocal identities: sin(theta) = 1/csc(theta), cos(theta) = 1/sec(theta), tan(theta) = 1/cot(theta)
- Quotient identities: tan(theta) = sin(theta)/cos(theta), cot(theta) = cos(theta)/sin(theta)
- Pythagorean identities: sin^2(theta) + cos^2(theta) = 1, 1 + tan^2(theta) = sec^2(theta), 1 + cot^2(theta) = csc^2(theta)
- Even-odd identities: sin(-theta) = -sin(theta), cos(-theta) = cos(theta), tan(-theta) = -tan(theta)
- Double angle identities: sin(2theta) = 2sin(theta)cos(theta), cos(2theta) = cos^2(theta) - sin^2(theta), tan(2theta) = 2tan(theta) / (1 - tan^2(theta))

These identities can be applied in various trigonometric problems to simplify calculations and find solutions.

### 5. Angle of Elevation and Depression

The angle of elevation is the angle between the horizontal line of sight and an upward line of sight from the observer to an object. The angle of depression is the angle between the horizontal line of sight and a downward line of sight from the observer to an object. These angles can be calculated using trigonometric ratios and are often used in real-world applications, such as surveying and navigation.

### 6. Law of Sines

The Law of Sines is a trigonometric law that relates the lengths of the sides of a triangle to the sines of its angles. It states that the ratio of the length of a side to the sine of its opposite angle is constant for all sides and angles of the triangle. The Law of Sines is expressed as:

a/sin(A) = b/sin(B) = c/sin(C)

Where a, b, and c are the lengths of the sides of the triangle, and A, B, and C are the measures of the angles opposite those sides.

The Law of Sines is useful in solving triangles when the lengths of two sides and the measure of one angle are known.

### 7. Law of Cosines

The Law of Cosines is a trigonometric law that relates the lengths of the sides of a triangle to the cosine of one of its angles. Unlike the Law of Sines, which relates the lengths of the sides to the sines of the angles, the Law of Cosines can be used to solve triangles when the lengths of all three sides are known. The Law of Cosines is expressed as:

c^2 = a^2 + b^2 - 2abcos(C)

Where c is the length of the side opposite angle C, and a and b are the lengths of the other two sides.

The Law of Cosines is useful in solving triangles when the lengths of all three sides or two sides and the included angle are known.

### 8. Unit Circle

The unit circle is a circle with a radius of 1 unit. It is used in trigonometry to define the values of trigonometric functions for all angles. The unit circle is divided into 360 degrees or 2π radians. The coordinates of points on the unit circle can be used to determine the values of sine, cosine, and tangent for different angles.

### 9. Trigonometric Equations

Trigonometric equations are equations that involve trigonometric functions. They can be solved using various techniques, such as factoring, substitution, or using trigonometric identities. Solving trigonometric equations requires a good understanding of the properties and relationships of trigonometric functions.

### 10. Trigonometric Inverses

Trigonometric inverses are functions that "undo" the trigonometric functions. They are denoted by adding a "-1" exponent to the function, such as sin^(-1), cos^(-1), and tan^(-1). Trigonometric inverses are used to find the angles corresponding to given trigonometric ratios.

### 11. Trigonometric Graphs

Trigonometric functions can be graphed to visualize their behavior and relationships. The graphs of sine, cosine, and tangent functions exhibit periodicity and symmetry. Understanding the graphs of trigonometric functions can help in analyzing and solving trigonometric equations and inequalities.

### 12. Applications of Trigonometry

Trigonometry has numerous applications in real-world situations. Some common applications include:

- Surveying: Trigonometry is used to measure distances, heights, and angles in surveying.
- Navigation: Trigonometry is used in navigation to determine positions, distances, and directions.
- Physics: Trigonometry is used in physics to analyze waveforms, oscillations, and motion.
- Engineering: Trigonometry is used in engineering to design structures, calculate forces, and analyze circuits.
- Astronomy: Trigonometry is used in astronomy to calculate distances, positions, and sizes of celestial objects.

These are just a few examples of how trigonometry is applied in various fields.

### Conclusion

Trigonometry is a fascinating branch of mathematics that has wide-ranging applications in many fields. Understanding the key concepts and formulas of trigonometry is essential for solving trigonometric problems and applications. By reviewing the topics covered in this article, you will be well-prepared for your Unit 8 Homework 6 assignment.