How to Use Special Triangles to Find Exact Values
Special triangles are a fundamental tool in trigonometry, providing a way to find exact values for trigonometric functions. These triangles, specifically the 30-60-90 triangle and the 45-45-90 triangle, have side lengths that are in a specific ratio, making them ideal for solving various trigonometric problems. In this article, we will explore how to use these special triangles to find exact values for sine, cosine, and tangent functions.
Understanding the Special Triangles
The first step in using special triangles to find exact values is to understand their properties. The 30-60-90 triangle has angles of 30 degrees, 60 degrees, and 90 degrees, with side lengths in the ratio of 1:√3:2. The 45-45-90 triangle, on the other hand, has angles of 45 degrees, 45 degrees, and 90 degrees, with side lengths in the ratio of 1:1:√2.
Using the 30-60-90 Triangle
The 30-60-90 triangle is particularly useful for finding exact values of sine and cosine for angles that are multiples of 30 degrees. For example, to find the exact value of sin(30°), we can use the side lengths of the 30-60-90 triangle. Since the sine of an angle is equal to the ratio of the opposite side to the hypotenuse, we have sin(30°) = opposite/hypotenuse = 1/2.
Similarly, to find the exact value of cos(30°), we use the adjacent side to the hypotenuse, resulting in cos(30°) = adjacent/hypotenuse = √3/2.
Using the 45-45-90 Triangle
The 45-45-90 triangle is useful for finding exact values of sine, cosine, and tangent for angles that are multiples of 45 degrees. For example, to find the exact value of sin(45°), we can use the side lengths of the 45-45-90 triangle. Since the sine of an angle is equal to the ratio of the opposite side to the hypotenuse, we have sin(45°) = opposite/hypotenuse = 1/√2.
To find the exact value of cos(45°), we use the adjacent side to the hypotenuse, resulting in cos(45°) = adjacent/hypotenuse = 1/√2.
For tangent, which is the ratio of the opposite side to the adjacent side, we have tan(45°) = opposite/adjacent = 1/1 = 1.
Combining Special Triangles
In some cases, we may need to use both special triangles to find exact values. For example, to find the exact value of sin(75°), we can break it down into sin(45° + 30°). Using the sum formula for sine, we have sin(75°) = sin(45°)cos(30°) + cos(45°)sin(30°). Substituting the known values, we get sin(75°) = (1/√2)(√3/2) + (1/√2)(1/2) = (√6 + √2)/4.
Conclusion
Using special triangles to find exact values is a valuable technique in trigonometry. By understanding the properties of the 30-60-90 and 45-45-90 triangles, we can quickly and accurately determine the exact values of sine, cosine, and tangent functions for angles that are multiples of 30 and 45 degrees. This knowledge can be applied to solve a variety of trigonometric problems and deepen our understanding of the subject.
