Trigonometry Signs: Solving For A, B, And C

Hey guys! Let's dive into a fun trigonometry problem. We're gonna figure out the signs of a, b, and c, which are defined by some trig functions. Don't worry, it's not as scary as it looks! We'll break it down step by step and make sure you understand every bit of it. Ready to roll?

Understanding the Problem: Signs of Trigonometric Functions

Alright, so here's the deal. We're given three values:

• a = sin(32π/3)
• b = cos(-π/3)
• c = tan(-1000)

Our mission is to figure out whether each of these values is positive (+), negative (-), or maybe even zero. This all boils down to understanding how sine, cosine, and tangent behave in different quadrants of the unit circle. Remember that unit circle? It's our best friend in trigonometry! The unit circle is a circle with a radius of 1 centered at the origin of a coordinate system. It is used to determine the trigonometric functions sine, cosine, and tangent. Let's start with sine, cosine, and tangent definitions. The sine of an angle is the y-coordinate of the point where the terminal side of the angle intersects the unit circle. The cosine of an angle is the x-coordinate of the point where the terminal side of the angle intersects the unit circle. The tangent of an angle is the ratio of the sine to the cosine of the angle (y/x).

Let's get into some detailed explanation here. Imagine the unit circle, split into four quadrants. In the first quadrant (0 to π/2 radians), both sine and cosine are positive. In the second quadrant (π/2 to π radians), sine is positive, but cosine is negative. In the third quadrant (π to 3π/2 radians), both sine and cosine are negative. Finally, in the fourth quadrant (3π/2 to 2π radians), sine is negative, but cosine is positive. The tangent is positive in the first and third quadrants (where sine and cosine have the same sign) and negative in the second and fourth quadrants (where sine and cosine have opposite signs). This will be useful when we get into solving the problem.

Analyzing 'a' = sin(32π/3)

First up, let's look at a = sin(32π/3). The sine function deals with angles, and the value of sine depends on the angle's position on the unit circle. The angle is 32π/3 radians. However, since the unit circle has a period of 2π, we can simplify this. We can subtract multiples of 2π until we get an angle between 0 and 2π. Let's do the math:

• 32π/3 - 2π = 26π/3
• 26π/3 - 2π = 20π/3
• 20π/3 - 2π = 14π/3
• 14π/3 - 2π = 8π/3
• 8π/3 - 2π = 2π/3

So, sin(32π/3) is the same as sin(2π/3). The angle 2π/3 is in the second quadrant. In the second quadrant, the sine function is positive. Therefore, a is positive (+).

Analyzing 'b' = cos(-π/3)

Now, let's consider b = cos(-π/3). The cosine function has a property that cos(-θ) = cos(θ). So, cos(-π/3) is the same as cos(π/3). The angle π/3 radians (which is 60 degrees) is in the first quadrant. In the first quadrant, the cosine function is positive. Thus, b is positive (+).

Analyzing 'c' = tan(-1000)

Finally, we'll look at c = tan(-1000). The tangent function has a period of π. So, we can add or subtract multiples of π to simplify the angle. Let's convert -1000 radians to an angle between 0 and 2π, or at least to a more manageable value. First, we need to know how many full rotations are in 1000 radians. Since 2π is a full rotation, we can divide 1000 by 2π to see how many rotations are there.

• 1000 / (2π) ≈ 159.155

This means that there are 159 full rotations. We can subtract these full rotations to get the equivalent angle. 159 full rotations are equivalent to 159 * 2π ≈ 999.512 radians. This means that -1000 radians is approximately -0.488 radians away from a full rotation. It's in the fourth quadrant. The tangent function is negative in the fourth quadrant. Therefore, c is negative (-). Alternatively, you can calculate the remainder when dividing -1000 by π and use this to determine the quadrant.

Putting It All Together: The Signs of a, b, and c

Alright, let's summarize our findings:

• a = sin(32π/3) is positive (+)
• b = cos(-π/3) is positive (+)
• c = tan(-1000) is negative (-)

Therefore, the correct answer is +, +, -. The question asks for the signs in the order a, b, and c.

Final Answer: Which Option is Correct?

So, if we look at the options provided, the one that matches our results is not listed. However, based on our calculations, the correct order is +, +, -. None of the options matches the answer we found, which is frustrating, to say the least! Let's double-check all of our calculations and make sure we didn't make any errors! It's always a good idea to double-check your work, guys. Mistakes happen! It appears there may be an error in the provided options. But hey, at least we know how to solve the problem!

Conclusion: Mastering Trig Signs

And that's a wrap, folks! We've successfully determined the signs of a, b, and c. Remember that the key is to understand the unit circle, the quadrants, and the behavior of sine, cosine, and tangent in each quadrant. Keep practicing, and you'll become a trigonometry whiz in no time. If you enjoyed this explanation, let me know. If you are struggling with a similar problem, feel free to ask. Cheers!