Understanding Trigonometric Functions at 390°: sin, cos, and tan

Introduction

In trigonometry, 390° is an important angle to understand because it is a rotation that completes a full circle of 360° plus an additional 30°. This angle helps us understand the periodic nature of trigonometric functions like sine, cosine, and tangent. Let's explore these values in depth.

What is 390°?

390° is one full rotation of 360° plus an additional 30°. When we reach 390°, we are actually at the same position as 30° due to the cyclical nature of trigonometric functions. This is because trigonometric functions have a period of 360°, meaning they repeat their values every 360°.

Trigonometric Functions at 390°

Since 390° is coterminal with 30°, we can use the values of 30° to find the values of sine, cosine, and tangent at 390°.

Sine

The sine of an angle is the y-coordinate of the corresponding point on the unit circle. The sine of 390° is the same as the sine of 30°.

sin 390° sin 30° 1/2

Cosine

The cosine of an angle is the x-coordinate of the corresponding point on the unit circle. The cosine of 390° is the same as the cosine of 30°.

cos 390° cos 30° √3/2

Tangent

The tangent of an angle is the sine of the angle divided by the cosine of the angle. Since both sine and cosine are positive, the tangent is also positive.

tan 390° sin 390° / cos 390° 1/2 / √3/2 1/√3 √3/3

Understanding the Values

When we rotate 360°, the trigonometric functions complete one full cycle and return to their initial values. Any additional rotation beyond 360° can be reduced to an angle between 0° and 360° by subtracting 360°. This is why 390° is equivalent to 30°.

Practical Examples

Imagine you are standing and looking at an object. If you turn 360°, you are back to the same position. Now, if you turn an additional 30° (390° in total), you are at the 30° angle. Here are the trigonometric values for 30°:

For trigonometric functions, these values are fundamental and should be memorized. Alternatively, you can use the properties of triangles, such as the 30°-60°-90° triangle and the 45°-45°-90° triangle. Remember to apply the Pythagorean theorem to further confirm these values.

Conclusion

Understanding the values of trigonometric functions at specific angles, such as 390°, is crucial for many applications in mathematics, physics, and engineering. By recognizing the periodic nature of these functions, you can simplify complex calculations and solve problems more efficiently.