Exact Values Of Trigonometric Expressions If Θ=π/6

When delving into trigonometry, understanding the values of trigonometric functions for specific angles is crucial. In this comprehensive guide, we will explore the exact values of various trigonometric expressions when θ = π/6. This angle, equivalent to 30 degrees, is a fundamental value in trigonometry, and mastering its trigonometric values is essential for solving a wide range of problems. We will break down the calculations step by step, ensuring a clear understanding of each concept. Whether you are a student learning trigonometry for the first time or someone looking to refresh your knowledge, this guide will provide you with the necessary tools to tackle trigonometric expressions with confidence.

(a) Finding the Exact Value of sin(-θ)

To find the exact value of sin(-θ) when θ = π/6, we need to understand the properties of the sine function. The sine function is an odd function, which means that sin(-x) = -sin(x) for any angle x. This property is crucial in simplifying trigonometric expressions involving negative angles. By applying this property, we can easily find the value of sin(-π/6). The sine function represents the y-coordinate of a point on the unit circle, and understanding its behavior in different quadrants helps in determining its value for various angles. In this section, we will explore the relationship between sine values of positive and negative angles and demonstrate how to apply the odd function property to solve the given expression. Let's dive into the step-by-step calculation and unravel the exact value of sin(-π/6).

Applying the Odd Function Property

As mentioned earlier, the sine function is an odd function. This means that for any angle x, the following holds true:

sin(-x) = -sin(x)

This property stems from the symmetry of the unit circle. When an angle is reflected across the x-axis, its sine value changes sign, while its cosine value remains the same. This is because the y-coordinate, which represents the sine value, becomes negative in the lower half of the unit circle. Understanding this fundamental property is crucial for simplifying trigonometric expressions involving negative angles. Now, let's apply this property to our specific case where θ = π/6.

Calculating sin(-π/6)

Given that θ = π/6, we want to find the value of sin(-θ), which is sin(-π/6). Using the odd function property, we can rewrite this as:

sin(-π/6) = -sin(π/6)

Now, we need to determine the value of sin(π/6). Recall that π/6 radians is equivalent to 30 degrees. The sine of 30 degrees is a well-known value in trigonometry, and it is equal to 1/2. This value can be derived from the 30-60-90 right triangle, where the side opposite the 30-degree angle is half the length of the hypotenuse. Therefore:

sin(π/6) = 1/2

Substituting this value back into our expression, we get:

sin(-π/6) = -sin(π/6) = -1/2

Thus, the exact value of sin(-θ) when θ = π/6 is -1/2. This result highlights the importance of understanding the properties of trigonometric functions and their values for standard angles. In summary, by applying the odd function property and knowing the sine value of π/6, we efficiently calculated sin(-π/6). This approach can be extended to other trigonometric functions and angles, providing a solid foundation for solving more complex problems.

(b) Determining the Exact Value of sin²θ

Next, we aim to find the exact value of sin²θ when θ = π/6. The notation sin²θ represents the square of the sine of θ, which is (sin θ)². This means we first need to find the value of sin θ and then square it. Understanding the concept of squaring trigonometric functions is crucial in many areas of mathematics and physics, including wave mechanics and signal processing. In this section, we will focus on the specific case of θ = π/6, but the principles discussed can be applied to any angle. We will revisit the value of sin(π/6) and demonstrate how to square it to obtain the final result. By mastering this simple yet fundamental operation, you will be better equipped to handle more complex trigonometric calculations. Let's proceed step by step to calculate the exact value of sin²(π/6).

Squaring the Sine Function

To calculate sin²θ, we first need to determine the value of sin θ and then square that value. Mathematically, this can be expressed as:

sin²θ = (sin θ)²

This notation is a shorthand way of representing the square of the sine function and is commonly used in trigonometric expressions. It's important to distinguish this from sin(θ²), which represents the sine of the angle θ squared. The order of operations matters significantly in these cases, and understanding the notation is crucial for accurate calculations. Now, let's apply this concept to our specific problem where θ = π/6.

Calculating sin²(π/6)

Given that θ = π/6, we want to find the value of sin²(π/6). From our previous discussion, we know that:

sin(π/6) = 1/2

Now, we need to square this value to find sin²(π/6). Squaring 1/2 is a straightforward arithmetic operation:

(1/2)² = (1/2) * (1/2) = 1/4

Therefore, the exact value of sin²(π/6) is 1/4. This result demonstrates the simple process of squaring a trigonometric function. By knowing the value of sin(π/6) and applying the squaring operation, we easily arrived at the final answer. This skill is essential for solving more complex trigonometric equations and problems. In summary, sin²(π/6) is calculated by first finding sin(π/6), which is 1/2, and then squaring it to get 1/4. This method can be applied to other trigonometric functions and angles, providing a solid foundation for further exploration in trigonometry.

(c) Evaluating the Exact Value of sin 2θ

Lastly, we will determine the exact value of sin 2θ when θ = π/6. This involves understanding the concept of the double angle and applying trigonometric identities. The double angle identity for sine is a crucial tool in simplifying and solving trigonometric expressions. It states that sin(2x) = 2sin(x)cos(x) for any angle x. This identity allows us to express the sine of twice an angle in terms of the sine and cosine of the angle itself. In this section, we will use this identity to find the value of sin(2π/6). We will also revisit the values of sin(π/6) and cos(π/6), which are necessary for applying the double angle identity. Let's proceed with the calculation, breaking it down into manageable steps to ensure a clear understanding. By the end of this section, you will be able to confidently evaluate the sine of double angles.

Understanding the Double Angle Identity

The double angle identity for sine is a fundamental trigonometric identity that relates the sine of twice an angle to the sine and cosine of the angle itself. The identity is expressed as:

sin(2x) = 2sin(x)cos(x)

This identity is derived from the angle sum identity for sine, which states that sin(A + B) = sin(A)cos(B) + cos(A)sin(B). By setting A = B = x, we can derive the double angle identity. Understanding the derivation of trigonometric identities helps in memorizing them and applying them correctly. The double angle identity is particularly useful in simplifying trigonometric expressions and solving equations involving double angles. Now, let's apply this identity to our specific case where θ = π/6.

Calculating sin(2π/6)

Given that θ = π/6, we want to find the value of sin(2θ), which is sin(2π/6). Using the double angle identity, we can rewrite this as:

sin(2π/6) = 2sin(π/6)cos(π/6)

Now, we need to determine the values of sin(π/6) and cos(π/6). We already know that:

sin(π/6) = 1/2

The cosine of π/6, which is cos(30°), is another well-known value in trigonometry. It can be derived from the 30-60-90 right triangle, where the side adjacent to the 30-degree angle is √3/2 times the length of the hypotenuse. Therefore:

cos(π/6) = √3/2

Substituting these values back into our expression, we get:

sin(2π/6) = 2 * (1/2) * (√3/2) = √3/2

Thus, the exact value of sin(2θ) when θ = π/6 is √3/2. This result demonstrates the application of the double angle identity in evaluating trigonometric expressions. By knowing the values of sin(π/6) and cos(π/6) and applying the identity, we efficiently calculated sin(2π/6). This approach can be extended to other trigonometric functions and angles, providing a solid foundation for solving more complex problems involving trigonometric identities. In summary, sin(2π/6) is calculated by applying the double angle identity, which requires knowing sin(π/6) and cos(π/6), and then substituting these values into the identity to obtain the final result.

In conclusion, we have successfully found the exact values of various trigonometric expressions when θ = π/6. These calculations demonstrate the importance of understanding the properties of trigonometric functions, including the odd function property, squaring trigonometric functions, and applying the double angle identity. By mastering these concepts and techniques, you will be well-equipped to tackle a wide range of trigonometric problems. Remember to practice regularly and apply these principles in different contexts to solidify your understanding. Happy trigonometry solving!