Trigonometric Ratios of Some Specific Angles

8.3 Trigonometric Ratios of Some Specific Angles

Description

Quick Overview

This section explores the values of trigonometric ratios for the angles of 0°, 30°, 45°, 60°, and 90° using geometric constructions.

Standard

Students will learn to calculate trigonometric ratios for specific angles like 0°, 30°, 45°, 60°, and 90°. This section derives these values through geometric constructions and the properties of right triangles, emphasizing the significance in trigonometry.

Detailed

Detailed Summary

In this section, we delve into the trigonometric ratios of key angles of 0°, 30°, 45°, 60°, and 90° through the application of right triangle properties and geometric constructions. We begin with calculating ratios for 45°, highlighting that in a right triangle where both angles are 45°, the legs are equal in length. By utilizing the Pythagorean theorem, we formulate the ratios, demonstrating that:
- sin 45° = cos 45° = √2/2
- tan 45° = 1

Next, we explore the ratios for angles 30° and 60° using an equilateral triangle. By drawing a perpendicular from one vertex, we create two right triangles, showing that:
- sin 30° = 1/2 and sin 60° = √3/2
- cos 30° = √3/2 and cos 60° = 1/2
- tan 30° = 1/√3 and tan 60° = √3

Finally, we address the limits of trigonometric functions as angles approach 0° and 90°, defining values for:
- sin 0° = 0, cos 0° = 1
- sin 90° = 1, cos 90° = 0

The section concludes by summarizing the values of all calculated trigonometric ratios in a convenient table for reference.

Key Concepts

  • Trigonometric Ratios: Defined for angles using sides of right triangles.

  • Sine of an Angle: Ratio of opposite side to hypotenuse.

  • Cosine of an Angle: Ratio of adjacent side to hypotenuse.

  • Values for Special Angles: Specific sine, cosine, and tangent values for 0°, 30°, 45°, 60°, and 90°.

Memory Aids

🎵 Rhymes Time

  • At 30° and 60° in trigonometry's dance, sin and cos take their chance.

📖 Fascinating Stories

  • Imagine a triangle where an angle shines bright at 45°. The pairs of sides represent equal delight!

🧠 Other Memory Gems

  • Acronym SOH-CAH-TOA: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent.

🎯 Super Acronyms

For 45° remember SCC

  • Sine
  • Cosine
  • Correspond equally.

Examples

  • Finding sin 30°: By constructing a right triangle, sin 30° = 1/2.

  • Calculating tan 45°: Using the definition, tan 45° = 1 since both opposing sides are equal.

Glossary of Terms

  • Term: Trigonometric Ratios

    Definition:

    Relational ratios of the sides of a right triangle, defining the sine, cosine, tangent, cosecant, secant, and cotangent functions.

  • Term: Sine

    Definition:

    Ratio of the length of the opposite side to the hypotenuse in a right triangle.

  • Term: Cosine

    Definition:

    Ratio of the length of the adjacent side to the hypotenuse in a right triangle.

  • Term: Tangent

    Definition:

    Ratio of the length of the opposite side to the adjacent side in a right triangle.

  • Term: Cosecant

    Definition:

    Reciprocal of sine, or 1/sin.

  • Term: Secant

    Definition:

    Reciprocal of cosine, or 1/cos.

  • Term: Cotangent

    Definition:

    Reciprocal of tangent, or 1/tan.