Let's break down this trigonometric problem step by step, guys. We need to evaluate the expression: sin(780°) + sin(120°) * cos(240°) + sin(390°). To do this, we'll first simplify each trigonometric function individually by finding their reference angles and then combine the results.

Step 1: Simplify sin(780°)

First, let's simplify sin(780°). Since trigonometric functions are periodic with a period of 360°, we can subtract multiples of 360° from 780° until we get an angle within the range of 0° to 360°.

780° - 360° = 420°

420° - 360° = 60°

So, sin(780°) = sin(60°). The sine of 60° is a well-known value, which is √3 / 2.

Therefore, sin(780°) = √3 / 2.

Step 2: Simplify sin(120°)

Next, let's simplify sin(120°). The angle 120° lies in the second quadrant. To find its reference angle, we subtract it from 180°:

180° - 120° = 60°

So, the reference angle for 120° is 60°. In the second quadrant, the sine function is positive. Therefore,

sin(120°) = sin(60°) = √3 / 2.

Step 3: Simplify cos(240°)

Now, let's simplify cos(240°). The angle 240° lies in the third quadrant. To find its reference angle, we subtract 180° from it:

240° - 180° = 60°

So, the reference angle for 240° is 60°. In the third quadrant, the cosine function is negative. Therefore,

cos(240°) = -cos(60°) = -1 / 2.

Step 4: Simplify sin(390°)

Finally, let's simplify sin(390°). Similar to the first step, we subtract multiples of 360° from 390° until we get an angle within the range of 0° to 360°.

390° - 360° = 30°

So, sin(390°) = sin(30°). The sine of 30° is a well-known value, which is 1 / 2.

Therefore, sin(390°) = 1 / 2.

Step 5: Combine the Results

Now that we have simplified each trigonometric function, we can substitute the values back into the original expression:

sin(780°) + sin(120°) * cos(240°) + sin(390°) = (√3 / 2) + (√3 / 2) * (-1 / 2) + (1 / 2)

Let's simplify this expression:

(√3 / 2) + (√3 / 2) * (-1 / 2) + (1 / 2) = (√3 / 2) - (√3 / 4) + (1 / 2)

To combine these terms, we need a common denominator, which is 4:

(2√3 / 4) - (√3 / 4) + (2 / 4) = (2√3 - √3 + 2) / 4

Combine the terms with √3:

(2√3 - √3 + 2) / 4 = (√3 + 2) / 4

So, the final result is:

(√3 + 2) / 4

Conclusion

Therefore, the value of the expression sin(780°) + sin(120°) * cos(240°) + sin(390°) is (√3 + 2) / 4. This process involves simplifying each trigonometric function using reference angles and periodicity, and then combining the results. Remember, guys, to always double-check the signs based on the quadrant in which the angle lies. This ensures accuracy in your calculations.

To truly master problems like these, it's essential to understand the underlying principles of trigonometric functions and how angles can be simplified. Trigonometric functions, such as sine, cosine, and tangent, relate the angles of a triangle to the ratios of its sides. These functions are periodic, meaning their values repeat at regular intervals. For sine and cosine, the period is 360 degrees (or 2π radians). This periodicity allows us to simplify angles by adding or subtracting multiples of 360 degrees without changing the value of the trigonometric function.

For instance, when we encountered sin(780°), we subtracted 360° twice to obtain sin(60°). This simplification is based on the property that sin(θ + 360n°) = sin(θ), where n is an integer. Understanding this concept is crucial for dealing with angles larger than 360 degrees.

Another important concept is the reference angle. The reference angle is the acute angle formed between the terminal side of the given angle and the x-axis. By using reference angles, we can determine the values of trigonometric functions for angles in any quadrant. The sign of the trigonometric function depends on the quadrant in which the angle lies. For example, in the second quadrant (90° < θ < 180°), sine is positive, while cosine and tangent are negative. In the third quadrant (180° < θ < 270°), tangent is positive, while sine and cosine are negative. And in the fourth quadrant (270° < θ < 360°), cosine is positive, while sine and tangent are negative.

When we simplified cos(240°), we found its reference angle to be 60°. Since 240° is in the third quadrant, where cosine is negative, we determined that cos(240°) = -cos(60°) = -1/2. These principles are fundamental for accurately evaluating trigonometric expressions.

When evaluating trigonometric expressions, it's easy to make mistakes if you're not careful. Here are some common pitfalls to watch out for:

1. Incorrectly Determining Reference Angles: Make sure you correctly identify the reference angle based on the quadrant in which the angle lies. For example, if an angle is in the second quadrant, subtract it from 180° to find the reference angle. If it's in the third quadrant, subtract 180° from the angle. And if it's in the fourth quadrant, subtract the angle from 360°.

2. Forgetting the Sign of the Trigonometric Function: Always remember to check the sign of the trigonometric function based on the quadrant. Use the mnemonic "All Students Take Calculus" (ASTC) to remember which functions are positive in each quadrant:

   • First Quadrant (All): All trigonometric functions are positive.
   • Second Quadrant (Students/Sine): Sine is positive.
   • Third Quadrant (Take/Tangent): Tangent is positive.
   • Fourth Quadrant (Calculus/Cosine): Cosine is positive.

3. Not Simplifying Angles Using Periodicity: If you're dealing with angles larger than 360°, make sure to subtract multiples of 360° to simplify them to angles within the range of 0° to 360°.

4. Using Incorrect Values for Special Angles: Memorize the values of trigonometric functions for common angles like 0°, 30°, 45°, 60°, and 90°. For example, sin(30°) = 1/2, cos(60°) = 1/2, and sin(45°) = cos(45°) = √2/2.

5. Arithmetic Errors: Double-check your arithmetic when combining the simplified trigonometric values. Make sure you have a common denominator when adding or subtracting fractions and that you correctly multiply and divide terms.

By being aware of these common mistakes and taking the time to double-check your work, you can improve your accuracy and avoid errors when evaluating trigonometric expressions.

To solidify your understanding, let's work through a few more practice problems:

1. Evaluate cos(690°) + tan(225°)

   • First, simplify cos(690°): 690° - 360° = 330°. So, cos(690°) = cos(330°). The reference angle for 330° is 360° - 330° = 30°. Since 330° is in the fourth quadrant, cosine is positive. Therefore, cos(330°) = cos(30°) = √3/2.
   • Next, simplify tan(225°): The reference angle for 225° is 225° - 180° = 45°. Since 225° is in the third quadrant, tangent is positive. Therefore, tan(225°) = tan(45°) = 1.
   • Combine the results: cos(690°) + tan(225°) = √3/2 + 1 = (√3 + 2)/2.

2. Evaluate sin(480°) - cos(570°)

   • First, simplify sin(480°): 480° - 360° = 120°. So, sin(480°) = sin(120°). The reference angle for 120° is 180° - 120° = 60°. Since 120° is in the second quadrant, sine is positive. Therefore, sin(120°) = sin(60°) = √3/2.
   • Next, simplify cos(570°): 570° - 360° = 210°. So, cos(570°) = cos(210°). The reference angle for 210° is 210° - 180° = 30°. Since 210° is in the third quadrant, cosine is negative. Therefore, cos(210°) = -cos(30°) = -√3/2.
   • Combine the results: sin(480°) - cos(570°) = √3/2 - (-√3/2) = √3/2 + √3/2 = √3.

By working through these practice problems, you'll gain confidence in your ability to evaluate trigonometric expressions and avoid common mistakes. Remember to always simplify angles, determine reference angles, check the signs of trigonometric functions, and double-check your arithmetic.

Evaluating trigonometric expressions can seem daunting at first, but by breaking down the problem into smaller steps and understanding the underlying principles, you can master this skill. Remember to simplify angles using periodicity, find reference angles, check the signs of trigonometric functions based on the quadrant, and avoid common mistakes. With practice and attention to detail, you'll be able to confidently evaluate even the most complex trigonometric expressions. So keep practicing, guys, and you'll become trigonometric masters in no time!