Mathematics 10C - Formula Sheet Guide for Trigonometry, Linear Functions, and Geometry

Mathematics 10C - Mathematics 10C is a foundational course that introduces key mathematical concepts such as measurement, trigonometry, linear functions, and geometric shapes. A formula sheet like the one provided serves as a quick reference to help students master complex problems efficiently. Whether you're solving for slope, surface area, or trigonometric ratios, understanding these formulas is critical for success in both academics and real-world applications.

This guide will break down the Mathematics 10C Formula Sheet to help you gain confidence and clarity while solving problems.

Measurement: Understanding Units

Imperial and Metric Systems

Measurement plays a critical role in Mathematics 10C. The two major systems of measurement are:

Imperial System – Used primarily in the U.S. and the UK.

• 1 ft = 12 in
• 1 yd = 3 ft
• 1 mi = 5280 ft

SI (Metric) System – The standard used worldwide.

• 1 in = 2.54 cm
• 1 ft = 0.3048 m
• 1 km = 0.6214 mi

Key Conversion Table

To solve real-world measurement problems, it's essential to convert between systems. Here’s an example:

Imperial to SI	SI to Imperial
1 mi = 1.6093 km	1 km = 0.6214 mi
1 yd = 0.9144 m	1 m = 1.0936 yds

Understanding these conversions is vital for geometry and surface area problems.

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Trigonometry Basics

Trigonometry revolves around the relationships between angles and sides in a right triangle.

• Hypotenuse: The longest side opposite the right angle.
• Opposite: The side across from the angle being referenced.
• Adjacent: The side next to the angle being referenced.

Pythagorean Theorem

The Pythagorean Theorem states:

$$a^2 + b^2 = c^2$$

Here:

• $a$ and $b$ are the legs of the triangle.
• $c$ is the hypotenuse.

Example: If $a = 3$ and $b = 4$, then $c = \sqrt{3^2 + 4^2} = 5$.

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Understanding the Pythagorean Theorem

The theorem is a powerful tool for solving unknown sides of right triangles. Steps to solve include:

1. Identify the hypotenuse.
2. Plug in the known side lengths into $a^2 + b^2 = c^2$.
3. Solve for the missing side.

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Trigonometric Ratios in Depth

Trigonometric ratios are the foundation of trigonometry:

Sine (sinθ):

$$\text{sinθ} = \frac{\text{opposite}}{\text{hypotenuse}}$$

Cosine (cosθ):

$$\text{cosθ} = \frac{\text{adjacent}}{\text{hypotenuse}}$$

Tangent (tanθ):

$$\text{tanθ} = \frac{\text{opposite}}{\text{adjacent}}$$

These ratios help calculate missing angles and sides in triangles.

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Linear Functions: Core Concepts

Linear functions represent straight-line relationships between two variables, $x$ and $y$. The formula for a line is:

$$y = mx + b$$

Where:

• $m$ = slope (rate of change)
• $b$ = y-intercept (starting point)

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Slope of a Line

The slope of a line determines its steepness. It can be calculated using the formula:

$$m = \frac{y_2 - y_1}{x_2 - x_1}$$

Here, $(x_1, y_1)$ and $(x_2, y_2)$ are two points on the line.

Example:

• If $(x_1, y_1) = (2, 3)$ and $(x_2, y_2) = (4, 7)$, then:

$$m = \frac{7 - 3}{4 - 2} = 2$$

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Slope-Intercept and Point-Slope Form

Slope-Intercept Form

The slope-intercept form is:

$$y = mx + b$$

Point-Slope Form

When you know the slope $m$ and one point $(x_1, y_1)$, use:

$$y - y_1 = m(x - x_1)$$

These forms are essential in graphing and solving equations.

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General Form of a Line

The general form is written as:

$$Ax + By + C = 0$$

It can be derived from slope-intercept form and is used in advanced problems.

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Parallel and Perpendicular Lines

Parallel Lines

Parallel lines have the same slope:

$$m_1 = m_2$$

Perpendicular Lines

Perpendicular slopes are negative reciprocals:

$$m_1 \times m_2 = -1$$

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Surface Area and Volume of 2-D Shapes

• Square: $$P = 4s, \, A = s^2$$
• Circle: $$C = 2\pi r, \, A = \pi r^2$$

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Surface Area and Volume of 3-D Objects

Formulas include:

Cube:

$$SA = 6s^2, \, V = s^3$$

Sphere:

$$SA = 4\pi r^2, \, V = \frac{4}{3} \pi r^3$$

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Exponents and Radicals

Key rules include:

$$a^m \cdot a^n = a^{m+n}, \quad (a^m)^n = a^{m \cdot n}, \quad a^0 = 1$$

Simplifying exponents and radicals is crucial for solving algebraic expressions.

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Practical Applications in Real Life

Trigonometry, measurement, and linear functions solve real-world challenges like navigation, construction, and physics problems.

Tips for Mastering Mathematics 10C

• Practice regularly.
• Memorize key formulas.
• Solve step-by-step.

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Frequently Asked Questions (FAQs)

1. What is the Pythagorean Theorem?

   • $a^2 + b^2 = c^2$.

2. How do you find the slope?

   • Use $m = \frac{y_2 - y_1}{x_2 - x_1}$.

3. What is surface area of a sphere?

• $SA = 4\pi r^2$.