Trigonometric Ratios of Specific Angles

Trigonometric ratios are fundamental concepts in trigonometry, relating the angles and sides of right-angled triangles. Specific angles like 0°, 30°, 45°, 60°, and 90° have well-defined and easily memorized trigonometric ratios. Understanding these ratios is crucial for solving various trigonometric problems and for building a solid foundation in mathematics. These angles are frequently encountered in geometry and physics.

Formulae

The basic trigonometric ratios are sine (sin), cosine (cos), and tangent (tan). Their values for the specific angles are as follows:

Angle (θ) | sin(θ) | cos(θ) | tan(θ)
0° | 0 | 1 | 0
30° | $ \frac{1}{2} $ | $ \frac{\sqrt{3}}{2} $ | $ \frac{1}{\sqrt{3}} $ or $ \frac{\sqrt{3}}{3} $
45° | $ \frac{\sqrt{2}}{2} $ | $ \frac{\sqrt{2}}{2} $ | 1
60° | $ \frac{\sqrt{3}}{2} $ | $ \frac{1}{2} $ | $ \sqrt{3} $
90° | 1 | 0 | Undefined (or ∞)

These values can be derived using the unit circle or by considering special right triangles (30-60-90 and 45-45-90). Remember: $tan(\theta) = \frac{sin(\theta)}{cos(\theta)}$

Examples

Example-1: Calculate the value of $sin(30°) + cos(60°)$.

Solution:
$sin(30°) = \frac{1}{2}$
$cos(60°) = \frac{1}{2}$
Therefore, $sin(30°) + cos(60°) = \frac{1}{2} + \frac{1}{2} = 1$

Example-2: Evaluate $2 * tan(45°) * sin(60°)$.

Solution:
$tan(45°) = 1$
$sin(60°) = \frac{\sqrt{3}}{2}$
Therefore, $2 * tan(45°) * sin(60°) = 2 * 1 * \frac{\sqrt{3}}{2} = \sqrt{3}$

Common mistakes by students

* Incorrect memorization of values: The most common mistake is forgetting the values of the trigonometric ratios for specific angles. Mnemonic devices and regular practice are vital. * Confusing sine and cosine: Students often mix up the sine and cosine values, especially for 30° and 60°. Always double-check which ratio corresponds to which angle. * Forgetting about the undefined value of tan(90°): Remember that tan(90°) is undefined because cos(90°) is 0, and division by zero is not allowed. * Using incorrect units: Ensure angles are expressed in degrees when using these specific ratios, or radians if that’s the context of the problem.

Real Life Application

Trigonometric ratios of specific angles are used in a wide variety of real-world scenarios:

Navigation: Determining distances and directions in sailing and aviation.
Engineering and Architecture: Calculating angles and dimensions in construction, bridge building, and structural design.
Surveying: Measuring land areas and elevations.
Physics: Analyzing forces and motion, particularly in projectile motion problems.
Computer Graphics: Creating 3D models and animations.

Fun Fact

You can create a simple table to remember the sine values for 0°, 30°, 45°, 60°, and 90°. Write 0, 1, 2, 3, and 4, divide each by 4, and take the square root. This gives you the values: $ \sqrt{\frac{0}{4}} , \sqrt{\frac{1}{4}} , \sqrt{\frac{2}{4}} , \sqrt{\frac{3}{4}} , \sqrt{\frac{4}{4}} $, which simplifies to 0, $ \frac{1}{2} $, $ \frac{\sqrt{2}}{2} $, $ \frac{\sqrt{3}}{2} $, and 1 – exactly the sine values! The cosine values are the same, but in reverse order.