How To Find Exact Value Of Trig Functions Given A Point
Trigonometry is an essential branch of mathematics that deals with the relationships between angles and sides of triangles. It has many practical applications in various fields such as engineering, physics, and navigation. One of the most fundamental concepts of trigonometry is the trigonometric functions, including sine, cosine, and tangent. In this article, we will discuss how to find the exact value of trig functions given a point.
Trigonometric Functions
Before we dive into finding the exact value of trig functions, let’s have a brief overview of trigonometric functions.
- The sine function (sin) is defined as the ratio of the opposite side to the hypotenuse of a right triangle.
- The cosine function (cos) is defined as the ratio of the adjacent side to the hypotenuse of a right triangle.
- The tangent function (tan) is defined as the ratio of the opposite side to the adjacent side of a right triangle.
These trig functions can be calculated for any angle, not only in the context of a right triangle. However, in this article, we will focus on finding the exact values of trig functions in a right triangle given a point.
Given Point
Let us consider a right triangle where the side lengths and angles are unknown. However, we are given a point on the triangle, and we are asked to find the exact value of trig functions at that point. For instance, let’s take a right triangle with an angle of 30 degrees and a hypotenuse of 2 units. Suppose the given point is on the opposite side of the angle, which is 1 unit away from the angle.
Finding the Side Lengths
To find the exact value of trig functions, we need to know the lengths of the sides of the triangle. We can use the Pythagorean theorem to calculate the unknown side lengths.
The Pythagorean theorem states that “the square of the hypotenuse is equal to the sum of the squares of the other two sides of a right triangle.”
In our example triangle, the hypotenuse is given as 2 units, and we know one side is 1 unit. Therefore, we can use the Pythagorean theorem to find the length of the other side:
c^2 = a^2 + b^2
2^2 = 1^2 + b^2
So, b^2 = 4 – 1 = 3
Therefore, b = square root of 3
Now, we know all three side lengths of the triangle. The opposite side from our angle measures 1 unit, the adjacent side measures the square root of 3 units, and the hypotenuse measures 2 units.
Exact Value of Trig Functions
To find the exact value of the trig functions, we need to use the ratios defined by those functions and the side lengths we just calculated.
The sine function (sin) is defined as the ratio of the opposite side to the hypotenuse of a right triangle.
sin(30) = opposite / hypotenuse = 1 / 2
The cosine function (cos) is defined as the ratio of the adjacent side to the hypotenuse of a right triangle.
cos(30) = adjacent / hypotenuse = sqrt(3) / 2
The tangent function (tan) is defined as the ratio of the opposite side to the adjacent side of a right triangle.
tan(30) = opposite / adjacent = 1 / sqrt(3)
Therefore, the exact value of sin(30) is 1/2, cos(30) is square root of 3 over 2, and tan(30) is 1 over square root of 3.
FAQs
What is the unit circle, and how is it used in trigonometry?
The unit circle is a circle with a radius of 1 unit, centered at the origin of a coordinate plane. It is used in trigonometry to define the values of trigonometric functions for all angles. The coordinates on the circle represent the values of sin and cos for the corresponding angle.
What are the reciprocal functions of trig functions?
The reciprocal functions of trig functions are secant, cosecant, and cotangent. They are defined as follows:
- The secant function (sec) is the reciprocal of the cosine function, i.e., sec(x) = 1/cos(x).
- The cosecant function (csc) is the reciprocal of the sine function, i.e., csc(x) = 1/sin(x).
- The cotangent function (cot) is the reciprocal of the tangent function, i.e., cot(x) = 1/tan(x).
How do we convert degrees to radians?
To convert degrees to radians, we use the formula:
radians = (degrees x pi) / 180
For example, 45 degrees is equal to (45 x pi) / 180, which simplifies to pi/4 radians.
What are the identities of trig functions?
The identities of trig functions are equations that are true for all values of the variables in a given domain. Some of the common identities are:
- The Pythagorean identity: sin^2(x) + cos^2(x) = 1.
- The reciprocal identity: sec(x) = 1/cos(x), csc(x) = 1/sin(x), and cot(x) = 1/tan(x).
- The quotient identity: tan(x) = sin(x) / cos(x) and cot(x) = cos(x) / sin(x).
What are the application areas of trigonometry?
Trigonometry has many practical applications in various fields. Some of the application areas are:
- Engineering: Trigonometry is used in the design and analysis of structures, machines, and systems.
- Physics: Trigonometry is used in the study of waves, motion, and other physical phenomena.
- Navigation: Trigonometry is used in navigation and positioning systems.
- Surveying: Trigonometry is used in surveying and land management.
Conclusion
Trigonometry is an essential branch of mathematics that has many practical applications. In this article, we discussed how to find the exact value of trig functions given a point. We learned how to calculate the lengths of the sides of a right triangle using the Pythagorean theorem and how to use those lengths to find the exact values of trig functions. Additionally, we discussed some of the frequently asked questions about trigonometry. We hope this article was helpful in understanding the basics of trigonometry.