Calculating Sin, Cos, Tan Values: A Step-by-Step Guide

Hey guys! Let's dive into the world of trigonometry and figure out how to calculate the sine, cosine, and tangent values for some angles. We'll be working with $1020^{\circ}$ and $-3\frac{1}{6}\pi$. Don't worry if it sounds intimidating; we'll break it down step by step to make it super clear and easy to understand. Ready to get started?

Understanding the Basics: Sine, Cosine, and Tangent

Before we jump into the calculations, let's refresh our memory on what sine, cosine, and tangent actually are. Think of the unit circle, that cool circle with a radius of 1. Angles are measured from the positive x-axis, and as you move around the circle, the sine, cosine, and tangent values change. Sine (sin) represents the y-coordinate of a point on the unit circle. Cosine (cos) represents the x-coordinate. And tangent (tan) is the ratio of sine to cosine (sin/cos). So basically, they're all about describing the relationship between angles and the sides of a right triangle or the coordinates on a circle. Understanding these fundamental concepts is key to mastering trigonometry, so if you're feeling a little shaky on these definitions, take a moment to review them. A strong foundation will make the rest of our calculations a breeze. Remember, practice makes perfect, so don't be afraid to try some examples on your own after we're done here. This will help you cement your understanding and boost your confidence in solving trigonometric problems. We'll use these definitions to calculate the values of the angles. The idea is to find the equivalent angle within the range of 0° to 360° (or 0 to 2π radians) because we know the sin, cos, and tan values for those angles.

The Unit Circle and Angle Measurement

We will be using the unit circle to calculate the sin, cos, and tan. The unit circle is a circle with a radius of 1 centered at the origin (0, 0) of a coordinate plane. An angle in the unit circle is formed by the positive x-axis and a line segment from the origin to a point on the circle. The angle's measurement is typically expressed in degrees or radians. One full revolution around the unit circle is 360 degrees or 2π radians. The unit circle is incredibly useful for understanding and visualizing trigonometric functions because the x-coordinate of the point where the line intersects the unit circle represents the cosine of the angle, and the y-coordinate represents the sine of the angle. Since the radius of the unit circle is 1, the sine and cosine values always fall between -1 and 1. The tangent of an angle is calculated as sine divided by cosine (tan θ = sin θ / cos θ). This helps determine the relationship between an angle and the ratios of the sides of a right triangle. The unit circle allows us to extend these trigonometric concepts to all angles, not just those within a right triangle, including angles greater than 90 degrees or even negative angles.

Key Trigonometric Identities

To make our calculations easier, it's helpful to remember some key trigonometric identities. These are basically equations that are always true, and they help us simplify complex expressions. One of the most important is the Pythagorean identity: sin² θ + cos² θ = 1. This comes directly from the unit circle, where the square of the x-coordinate (cosine) plus the square of the y-coordinate (sine) equals the square of the radius, which is 1. Another useful set of identities involves the relationship between an angle and its negative counterpart. For example, sin(-θ) = -sin θ, cos(-θ) = cos θ, and tan(-θ) = -tan θ. These identities are important when we deal with negative angles, like in our problem. The periodic identities are also essential. Because the trigonometric functions repeat themselves in cycles, we can simplify an angle like $1020^{\circ}$ by finding its equivalent within a single cycle (0° to 360°). The sine, cosine, and tangent functions have periods of 360° (or 2π radians), which means their values repeat every 360 degrees. This will be very useful when we calculate the angles given. Make sure you're familiar with these identities; they'll save you a lot of time and effort in trigonometry!

Calculation of $1020^{\circ}$

Alright, let's tackle our first angle: $1020^{\circ}$. This is a pretty big angle, but don't worry, we can simplify it. The trick here is to find an angle between 0° and 360° that has the same trigonometric values. We do this by subtracting multiples of 360° (a full circle) until we get an angle within that range. Let's do it! Since a full circle is 360 degrees, we can subtract multiples of 360 from 1020 until we get a value between 0 and 360. $1020^{\circ} - 360^{\circ} = 660^{\circ}$. But 660° is still greater than 360°. So, let's subtract another 360°: $660^{\circ} - 360^{\circ} = 300^{\circ}$. Bingo! $300^{\circ}$ is between 0° and 360°. That means $1020^{\circ}$ is the same as $300^{\circ}$ in terms of sine, cosine, and tangent. Now, we just need to find the sin, cos, and tan of $300^{\circ}$.

Finding the Reference Angle and Quadrant

Next, we need to determine the reference angle. This is the acute angle formed between the terminal side of the angle and the x-axis. For an angle of $300^{\circ}$, which lies in the fourth quadrant, the reference angle is $360^{\circ} - 300^{\circ} = 60^{\circ}$. Now, the quadrant is important because it tells us the signs of the trigonometric functions. In the fourth quadrant, cosine is positive, and sine and tangent are negative. Remember the mnemonic