Trigonometri Kelas 10 Kurikulum Merdeka - Rumus Lengkap - Penjelasan mudah dipahami - Langsung Bisa

Introduction

This tutorial provides a comprehensive overview of trigonometry concepts covered in the 10th-grade curriculum, specifically tailored for the Merdeka curriculum. It includes essential formulas, explanations of special angles, and applications of trigonometry in various quadrants. This guide aims to simplify the learning process, making it accessible and easy to understand.

Step 1: Understanding Right Triangle Trigonometry

• Basic Definitions:

  • Trigonometry deals with the relationships between the angles and sides of triangles, particularly right triangles.

• Key Ratios:

  • Sine (sin): Ratio of the length of the opposite side to the hypotenuse.
    • Formula: ( \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} )
  • Cosine (cos): Ratio of the length of the adjacent side to the hypotenuse.
    • Formula: ( \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} )
  • Tangent (tan): Ratio of the length of the opposite side to the adjacent side.
    • Formula: ( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} )

• Practical Advice:

  • Always identify the right triangle and label the sides according to the angle of interest before applying the formulas.

Step 2: Special Angle Ratios

• Common Angles:

  • Memorize the sine, cosine, and tangent values for special angles (0°, 30°, 45°, 60°, and 90°).

• Values:

  • ( \sin(0°) = 0 )
  • ( \sin(30°) = \frac{1}{2} )
  • ( \sin(45°) = \frac{\sqrt{2}}{2} )
  • ( \sin(60°) = \frac{\sqrt{3}}{2} )
  • ( \sin(90°) = 1 )

• Practical Tip:

  • Use a unit circle to visualize these angles and their corresponding sine and cosine values.

Step 3: Trigonometric Ratios in All Quadrants

• Understanding Quadrants:

  • The unit circle is divided into four quadrants, each affecting the sign of the trigonometric ratios.

• Sign Rules:

  • Quadrant I: All ratios are positive.
  • Quadrant II: Sine is positive, cosine and tangent are negative.
  • Quadrant III: Tangent is positive, sine and cosine are negative.
  • Quadrant IV: Cosine is positive, sine and tangent are negative.

• Practical Advice:

  • Determine the quadrant of the angle to correctly evaluate the trigonometric functions.

Step 4: Negative Angles and Angles Over 360 Degrees

• Negative Angles:

  • The value of trigonometric functions for negative angles can be found using the following:
    • ( \sin(-\theta) = -\sin(\theta) )
    • ( \cos(-\theta) = \cos(\theta) )
    • ( \tan(-\theta) = -\tan(\theta) )

• Angles Greater than 360 Degrees:

  • For angles greater than 360°, subtract 360° until the angle is within the 0° to 360° range.

• Practical Tip:

  • Use reference angles to simplify calculations for both negative angles and angles over 360°.

Step 5: Equations and Polar Coordinates

• Trigonometric Equations:

  • Familiarize yourself with basic equations involving trigonometric functions and learn how to solve them.

• Polar and Cartesian Coordinates:

  • Understand how to convert between polar coordinates (r, θ) and Cartesian coordinates (x, y) using:
    • ( x = r \cos(\theta) )
    • ( y = r \sin(\theta) )

• Practical Application:

  • Knowing how to convert and calculate in different coordinate systems is essential for solving real-world problems.

Conclusion

This guide covers essential trigonometric concepts, including right triangle ratios, special angle values, trigonometric functions in different quadrants, the treatment of negative angles, and conversions between coordinate systems. By mastering these topics, you will build a solid foundation in trigonometry. For further practice, consider solving problems or exploring more advanced trigonometric identities and applications.