Learn the basic trigonometric ratios and their relationships.
Example 1: Introduction to Trigonometry
Understand what trigonometry is and its focus.
Definition: Trigonometry is the branch of mathematics that deals with the relationships between the sides and angles of triangles, particularly right triangles.
Focus: It helps us find unknown sides or angles of a right triangle when some information is given.
Key Concept: For a specific angle in a right triangle, the ratio of the lengths of two sides is constant, regardless of the size of the triangle. These ratios are called trigonometric ratios.
Example 2: Trigonometric Ratios (Sine, Cosine, Tangent)
Define the primary trigonometric ratios for an acute angle in a right triangle.
Consider a Right Triangle ABC, with right angle at B. Let's focus on acute angle A.
**Opposite Side:** The side opposite to angle A (side BC).
**Adjacent Side:** The side adjacent to angle A (side AB).
**Hypotenuse:** The side opposite the right angle (side AC). This is always the longest side.
Sine of Angle A (sin A): Ratio of the length of the side opposite to angle A to the length of the hypotenuse.
sin A = Opposite Side / Hypotenuse = BC / AC
Cosine of Angle A (cos A): Ratio of the length of the side adjacent to angle A to the length of the hypotenuse.
cos A = Adjacent Side / Hypotenuse = AB / AC
Tangent of Angle A (tan A): Ratio of the length of the side opposite to angle A to the length of the side adjacent to angle A.
tan A = Opposite Side / Adjacent Side = BC / AB
Mnemonic: SOH CAH TOA can help remember these ratios:
SOH: **S**in = **O**pposite / **H**ypotenuse
CAH: **C**os = **A**djacent / **H**ypotenuse
TOA: **T**an = **O**pposite / **A**djacent
Example 3: Reciprocal Relationships
Understand the reciprocal trigonometric ratios.
Cosecant (csc A): The reciprocal of sine A.
csc A = 1 / sin A = Hypotenuse / Opposite Side
Secant (sec A): The reciprocal of cosine A.
sec A = 1 / cos A = Hypotenuse / Adjacent Side
Cotangent (cot A): The reciprocal of tangent A.
cot A = 1 / tan A = Adjacent Side / Opposite Side
Note: These reciprocal ratios are defined only when the denominator is not zero.
Example 4: Relationship between Tan, Sin, and Cos
Understand that tangent can be expressed in terms of sine and cosine.
Relationship: For any acute angle A, tan A is equal to the ratio of sin A to cos A.
tan A = sin A / cos A
Proof: We know sin A = Opposite / Hypotenuse
We know cos A = Adjacent / Hypotenuse
Divide sin A by cos A: (Opposite / Hypotenuse) / (Adjacent / Hypotenuse)
Simplify the division: (Opposite / Hypotenuse) * (Hypotenuse / Adjacent)
The 'Hypotenuse' terms cancel out.
Result: Opposite / Adjacent
This is the definition of tan A.
Therefore, tan A = sin A / cos A.
Practice Mode - Trigonometric Ratio Calculator
Enter the lengths of the sides of a right triangle (relative to an acute angle) to calculate the basic trigonometric ratios.
Note: Assume the angle is 'theta'. Enter positive numbers for side lengths.
Related Concepts
Explore these related mathematical concepts.
Right Triangle
A triangle with one 90-degree angle.
Hypotenuse
The side opposite the right angle.
Opposite Side
The side across from a given angle.
Adjacent Side
The side next to a given angle (not the hypotenuse).
Angle
Formed by two rays sharing a common endpoint.
Ratio
A comparison of two quantities by division.
Pythagorean Theorem
Relates the sides of a right triangle (a^2 + b^2 = c^2).
Trigonometric Identity
An equation involving trig ratios that is true for all angles.