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Grade 10 Trigonometry

Interactive step-by-step solver for understanding trigonometric ratios, identities, and applications.

θ Adjacent Opposite Hypotenuse
Back to Grade 10 Chapter 11 Chapter 13
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Step-by-Step Learning

Learn the fundamentals of trigonometry, focusing on right-angled triangles.

Example 1: Basic Trigonometric Ratios (SOH CAH TOA)

Explain the definitions of Sine, Cosine, and Tangent in a right-angled triangle.

Introduction: Trigonometry deals with the relationships between the sides and angles of triangles. For Grade 10, we focus on right-angled triangles.
Sides of a Right Triangle: Relative to a non-right angle (let's call it $ \theta $):
  • **Hypotenuse:** The side opposite the right angle (always the longest side).
  • **Opposite Side:** The side directly across from angle $ \theta $.
  • **Adjacent Side:** The side next to angle $ \theta $ (not the hypotenuse).
Trigonometric Ratios: These are ratios of the lengths of the sides of a right triangle.
  • **Sine (sin):** $ \text{sin}(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}} $
  • **Cosine (cos):** $ \text{cos}(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} $
  • **Tangent (tan):** $ \text{tan}(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} $
Mnemonic: A common way to remember these ratios is **SOH CAH TOA**:
  • **S**in = **O**pposite / **H**ypotenuse
  • **C**os = **A**djacent / **H**ypotenuse
  • **T**an = **O**pposite / **A**djacent
Basic Ratios θ Adjacent Opposite Hypotenuse sin(θ) = Opp/Hyp cos(θ) = Adj/Hyp tan(θ) = Opp/Adj

Example 2: Reciprocal Ratios

Explain the definitions of Cosecant, Secant, and Cotangent.

Reciprocal Ratios: In addition to Sine, Cosine, and Tangent, there are three other trigonometric ratios which are the reciprocals of the basic ones.
Cosecant (csc or cosec): The reciprocal of Sine.
$ \text{csc}(\theta) = \frac{1}{\text{sin}(\theta)} = \frac{\text{Hypotenuse}}{\text{Opposite}} $
Secant (sec): The reciprocal of Cosine.
$ \text{sec}(\theta) = \frac{1}{\text{cos}(\theta)} = \frac{\text{Hypotenuse}}{\text{Adjacent}} $
Cotangent (cot): The reciprocal of Tangent.
$ \text{cot}(\theta) = \frac{1}{\text{tan}(\theta)} = \frac{\text{Adjacent}}{\text{Opposite}} $
Reciprocal Ratios csc(θ) = 1 / sin(θ) sec(θ) = 1 / cos(θ) cot(θ) = 1 / tan(θ) These are the inverses of the basic ratios

Example 3: Trigonometric Identities

Explain a fundamental trigonometric identity, like the Pythagorean Identity.

Trigonometric Identities: Equations that are true for all values of the variables for which both sides of the equation are defined. They are useful for simplifying expressions and solving equations.
Pythagorean Identity: One of the most important identities, derived from the Pythagorean theorem ($ a^2 + b^2 = c^2 $).
$ \text{sin}^2(\theta) + \text{cos}^2(\theta) = 1 $
Other Identities: There are many other identities, including:
  • $ \text{tan}(\theta) = \frac{\text{sin}(\theta)}{\text{cos}(\theta)} $
  • $ 1 + \text{tan}^2(\theta) = \text{sec}^2(\theta) $
  • $ 1 + \text{cot}^2(\theta) = \text{csc}^2(\theta) $
Pythagorean Identity sin²(θ) + cos²(θ) = 1 Derived from the Pythagorean Theorem

Example 4: Angles of Elevation and Depression

Explain the concepts of angle of elevation and angle of depression.

Line of Sight: The imaginary line from an observer's eye to the object being viewed.
Horizontal Line: An imaginary line extending horizontally from the observer's eye.
Angle of Elevation: The angle formed by the horizontal line and the line of sight **when looking upwards** at an object.
Angle of Depression: The angle formed by the horizontal line and the line of sight **when looking downwards** at an object.
Note: The angle of elevation from point A to point B is equal to the angle of depression from point B to point A (assuming A and B are at different heights), because they are alternate interior angles formed by parallel horizontal lines.
Angles of Elevation/Depression Horizontal Line Line of Sight (Up) Elevation Line of Sight (Down) Depression

Practice Mode

Enter a simple trigonometry problem to solve. This basic solver can help with calculating ratios or finding missing sides/angles in right triangles.

Note: Enter problems like "sin(30)", "tan(45 degrees)", "find hypotenuse if opposite=3, angle=30 degrees", "find angle if opposite=4, adjacent=3". Specify angles in degrees.

Related Concepts

Explore these related mathematical concepts to deepen your understanding of trigonometry.

Right Triangle

A triangle with one angle measuring 90 degrees.

Hypotenuse

The side opposite the right angle in a right triangle.

Opposite Side

The side across from a given angle in a right triangle.

Adjacent Side

The side next to a given angle in a right triangle (not the hypotenuse).

SOH CAH TOA

A mnemonic to remember the basic trigonometric ratios (Sine, Cosine, Tangent).

Trigonometric Identities

Equations involving trigonometric functions that are true for all valid inputs.

Angle of Elevation

The angle upwards from a horizontal line to a line of sight.

Angle of Depression

The angle downwards from a horizontal line to a line of sight.