Cara Menentukan Nilai Perbandingan Trigonometri Sudut Lebih Dari 360 Derajat Lebih Dari Satu Putaran

Summary

This video tutorial explains the concept of trigonometric angles greater than 360°, focusing on how to calculate the values of trigonometric functions such as sine and cosine for these angles. It introduces two key relations for handling angles beyond 360°: subtracting multiples of 360° or adding them to find equivalent angles within one rotation. Through examples, it demonstrates how to apply these relations in different quadrants and how the signs of trigonometric functions change based on the quadrant. The tutorial also walks through several practice problems to reinforce the concepts.

Takeaways

• 😀 Angles greater than 360° indicate multiple rotations, and their values can be calculated by subtracting or adding multiples of 360°.
• 😀 A single full rotation in trigonometry is 360°, so for angles greater than 360°, subtracting or adding multiples of 360° helps in determining equivalent angles.
• 😀 The trigonometric functions (sine, cosine, tangent) are dependent on which quadrant the terminal side of the angle lies in.
• 😀 For angles greater than 360°, you can use formulas like n × 360° ± alpha, where n is the number of rotations and alpha is the remaining angle.
• 😀 For example, for an angle like 660°, subtracting multiples of 360° (e.g., 720° - 660°) gives an equivalent smaller angle like 60°.
• 😀 Angles in different quadrants have different signs for their trigonometric values. In quadrant 1, all functions are positive; in quadrant 4, sine is negative, while cosine and tangent are positive.
• 😀 When using the angle formula n × 360° ± alpha, one can determine which quadrant the angle’s terminal side will be in, helping to identify the correct sign for the trigonometric function.
• 😀 To determine trigonometric values, it’s important to reduce large angles to a manageable size by subtracting or adding multiples of 360°, then calculate the standard trigonometric value.
• 😀 For example, for 1125°, subtracting 1080° (3 × 360°) results in an angle of 45°, which can be used to calculate the trigonometric value (cos 45°).
• 😀 The angle reduction technique can be used for angles larger than 1000°, 3000°, etc., where multiple rotations are subtracted to bring the angle into a standard range (0° to 360°).

Q & A

What is the concept of angles greater than 360° in trigonometry?

Angles greater than 360° represent more than one complete rotation. When dealing with such angles, we need to break them down by subtracting multiples of 360° to bring them into the standard range of 0° to 360° for easier calculation of trigonometric functions.

How do we calculate the sine of an angle greater than 360°?

To calculate the sine of an angle greater than 360°, we subtract multiples of 360° from the angle until we bring it within the 0° to 360° range. For example, for an angle of 660°, we subtract 360° twice (720° - 660° = 60°), and then calculate sine 60°.

What happens to the sign of the sine function in different quadrants?

In quadrant 1, sine is positive. In quadrant 2, sine is positive, but cosine and tangent are negative. In quadrant 3, sine is negative, and in quadrant 4, sine is negative as well.

Why do we use two relations (n*360° ± alpha) when working with angles over 360°?

We use two relations to account for multiple rotations. The relation n*360° ± alpha helps to simplify the angle and bring it into the standard range, allowing us to calculate the sine or cosine of the angle.

In the example with 660°, why is sine negative?

In this case, 660° is equivalent to 60° in quadrant 4 after reducing the angle by subtracting 720° - 660°. Since sine is negative in quadrant 4, the value of sine 660° is negative.

How does cosine behave for angles greater than 360°?

Cosine values for angles greater than 360° are calculated by subtracting multiples of 360° and then determining which quadrant the final angle lies in. For example, with 1125°, we subtract 3*360° (1080°) and then calculate cosine of the remaining 45°.

How can we simplify the angle 1125° to find its cosine?

To simplify 1125°, we subtract 3*360° (1080°), leaving us with an angle of 45°. Since cosine is positive in quadrant 1, we can directly calculate cos 45°.

What is the role of the quadrants when calculating trigonometric functions for large angles?

The quadrants determine the sign of trigonometric functions. For example, sine and cosine have different signs depending on which quadrant the angle ends up in after reducing the angle. This helps in determining whether the final trigonometric value is positive or negative.

What should we do if we have an angle greater than 1000°, like 1111°?

For an angle like 1111°, we subtract multiples of 360° to reduce it. In this case, 1111° - 1080° = 31°, and then we calculate sin 31° because 31° is within the 0° to 360° range.

Why is the answer for sin 1111° equivalent to sin 31°?

After reducing 1111° by subtracting 1080°, we are left with 31°, which lies in quadrant 1 where sine is positive. Therefore, sin 1111° is equal to sin 31°.