The sine, cosine, and tangent functions are important mathematical tools that are used in a variety of fields, including physics, engineering, and mathematics. These functions are part of the larger family of trigonometric functions and are defined in terms of the right-angled triangle sides. In this article, we will discuss how to apply the sine, cosine, and tangent functions to solve problems in trigonometry.

Definition

First, let’s define the sine, cosine, and tangent functions. In a right-angled triangle, the sine function is described as the ratio of the side opposite to an angle’s length to the hypotenuse’s length. The cosine function is described as the ratio of the adjacent side’s length to the hypotenuse’s length. Finally, the tangent function is described as the ratio of the opposite side’s length to the adjacent side’s length.

Application

To apply these functions, we first need to identify the angle we are interested in and the triangle sides that are relevant to that angle. For example, if we are interested in the angle θ, we need to identify the side opposite to θ, the side adjacent to θ, and the hypotenuse. We can then use the definitions of the sine, cosine, and tangent functions to find the values of these functions for the given angle.

Let’s consider an example problem to demonstrate how to apply these functions. Suppose we have a right-angled triangle with a hypotenuse of length ten and an angle θ of 30 degrees. We want to find the length of the side opposite to θ and the length of the side adjacent to θ.

First, one will be able to employ the definition of the sine function to find the length of the side opposite to θ. The sine of θ is described as the ratio of the opposite side’s length to the hypotenuse’s

length. Therefore, sin(θ) = opposite/hypotenuse. We know the value of the hypotenuse is 10, and the value of θ is 30 degrees, so we can substitute these values into the formula to get sin(30) = opposite/10. If we solve for the opposite side, we will get opposite = 10*sin(30) = 5.

Next, we can use the definition of the cosine function to find the length of the side adjacent to θ. The cosine of θ is described as the ratio of the adjacent side’s length to the hypotenuse’s length. Therefore, cos(θ) = adjacent/hypotenuse. We know the value of the hypotenuse is 10, and the value of θ is 30 degrees, so we can substitute these values into the formula to get cos(30) = adjacent/10. Solving for the adjacent side, we get adjacent = 10*cos(30) = 8.66.

Finally, we can use the definition of the tangent function to search for the value of the tangent of θ. The tangent of θ is described as the ratio of the length’s opposite side to the adjacent side’s length. Hence, tan(θ) = adjacent/opposite. We already know the values of the opposite and adjacent sides, so we can substitute them into the formula to get tan(30) = 5/8.66 = 0.58.

Conclusion

To apply the sine, cosine, and tangent functions, we first need to identify the angle we are interested in and the triangle’s sides that are relevant to that angle. We can then use the definitions of these functions to search for the values of these functions for the given angle. The sine function gives the ratio of the length of the side that is opposite to the angle.

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