Trigonometric ratios
Understanding Trigonometric Ratios
Trigonometric ratios are fundamental to the study of trigonometry, which is a branch of mathematics that deals with the relationships between the sides and angles of triangles, particularly right-angled triangles. These ratios are used to describe the relationship between the angles and lengths of the sides of a triangle.
Basic Trigonometric Ratios
There are six basic trigonometric ratios, which are defined for acute angles as follows:
- Sine (sin): The ratio of the length of the opposite side to the length of the hypotenuse.
- Cosine (cos): The ratio of the length of the adjacent side to the length of the hypotenuse.
- Tangent (tan): The ratio of the length of the opposite side to the length of the adjacent side.
- Cosecant (csc): The reciprocal of sine, or the ratio of the length of the hypotenuse to the length of the opposite side.
- Secant (sec): The reciprocal of cosine, or the ratio of the length of the hypotenuse to the length of the adjacent side.
- Cotangent (cot): The reciprocal of tangent, or the ratio of the length of the adjacent side to the length of the opposite side.
Trigonometric Ratios Formulas
For a right-angled triangle with an angle ( \theta ), the sides can be defined as follows:
- Opposite (O): The side opposite to the angle ( \theta )
- Adjacent (A): The side next to the angle ( \theta ) and perpendicular to the hypotenuse
- Hypotenuse (H): The side opposite the right angle, and the longest side of the triangle
The trigonometric ratios can be expressed using the following formulas:
- ( \sin(\theta) = \frac{O}{H} )
- ( \cos(\theta) = \frac{A}{H} )
- ( \tan(\theta) = \frac{O}{A} )
- ( \csc(\theta) = \frac{1}{\sin(\theta)} = \frac{H}{O} )
- ( \sec(\theta) = \frac{1}{\cos(\theta)} = \frac{H}{A} )
- ( \cot(\theta) = \frac{1}{\tan(\theta)} = \frac{A}{O} )
Table of Trigonometric Ratios
| Ratio | Definition | Reciprocal |
|---|---|---|
| sin | ( \frac{O}{H} ) | csc |
| cos | ( \frac{A}{H} ) | sec |
| tan | ( \frac{O}{A} ) | cot |
| csc | ( \frac{H}{O} ) | sin |
| sec | ( \frac{H}{A} ) | cos |
| cot | ( \frac{A}{O} ) | tan |
Examples to Explain Important Points
Example 1: Calculating Trigonometric Ratios
Suppose we have a right-angled triangle with an acute angle ( \theta ), where the opposite side is 3 units long, the adjacent side is 4 units long, and the hypotenuse is 5 units long. The trigonometric ratios for angle ( \theta ) would be:
- ( \sin(\theta) = \frac{3}{5} )
- ( \cos(\theta) = \frac{4}{5} )
- ( \tan(\theta) = \frac{3}{4} )
- ( \csc(\theta) = \frac{5}{3} )
- ( \sec(\theta) = \frac{5}{4} )
- ( \cot(\theta) = \frac{4}{3} )
Example 2: Using Trigonometric Ratios to Find Unknown Sides
Imagine you know the angle ( \theta = 30^\circ ) and the hypotenuse ( H = 10 ) units of a right-angled triangle. You can use the trigonometric ratios to find the lengths of the opposite and adjacent sides.
- Using the sine ratio: ( \sin(30^\circ) = \frac{1}{2} ), so the opposite side ( O = \sin(30^\circ) \times H = \frac{1}{2} \times 10 = 5 ) units.
- Using the cosine ratio: ( \cos(30^\circ) = \frac{\sqrt{3}}{2} ), so the adjacent side ( A = \cos(30^\circ) \times H = \frac{\sqrt{3}}{2} \times 10 \approx 8.66 ) units.
Example 3: Solving for an Angle Using Trigonometric Ratios
If you know the lengths of the sides of a right-angled triangle, you can solve for the angles using the inverse trigonometric functions. For instance, if the opposite side is 7 units and the hypotenuse is 25 units, you can find angle ( \theta ) as follows:
- ( \sin(\theta) = \frac{7}{25} )
- ( \theta = \sin^{-1}\left(\frac{7}{25}\right) \approx 16.26^\circ )
Important Points to Remember
- Trigonometric ratios are only defined for angles between ( 0^\circ ) and ( 90^\circ ) in a right-angled triangle.
- The values of the trigonometric ratios for specific angles (like ( 0^\circ ), ( 30^\circ ), ( 45^\circ ), ( 60^\circ ), and ( 90^\circ )) are often used and should be memorized.
- The reciprocal trigonometric ratios (csc, sec, and cot) are not as commonly used as sin, cos, and tan, but they are important for solving certain types of problems.
- Understanding the unit circle can help visualize and remember the values of trigonometric ratios for various angles.
By mastering the trigonometric ratios and their applications, you will be well-equipped to tackle a wide range of problems in trigonometry, from simple calculations to more complex applications in geometry, physics, engineering, and beyond.