The sine function is a fundamental trigonometric function that represents the ratio of the length of the side opposite an angle to the hypotenuse in a right triangle. While it is easy to find the sine of common angles such as 30, 45, or 60 degrees, what about less common angles like 105 degrees? In this article, we will explore a method to determine the exact value of sine 105 degrees.
Finding the exact value of sine 105 degrees requires the use of trigonometric identities and the knowledge of common trigonometric values. One crucial identity we will be using is the sum to product identity, which states that sin(A + B) = sin(A)cos(B) + cos(A)sin(B).
To begin, we can rewrite 105 degrees as the sum of two angles that yield known sine values. Let’s express it as 90 degrees + 15 degrees. Therefore, sin 105 degrees can be written as sin(90 degrees + 15 degrees).
By applying the sum to product identity, we can rewrite sin 105 degrees as sin(90 degrees)cos(15 degrees) + cos(90 degrees)sin(15 degrees).
Since sin 90 degrees equals 1 and cos 90 degrees equals 0, we can simplify the equation to sin 105 degrees = 1 * cos(15 degrees) + 0 * sin(15 degrees).
Now, let’s focus on evaluating cos 15 degrees and sin 15 degrees individually. Unfortunately, these values are not commonly found on trigonometric tables. However, we can use another trigonometric identity, the half-angle formula, to find their exact values.
The half-angle formula states that cos(x/2) = ±√[(1 + cos x)/2]. By plugging in x = 30 degrees, we can determine the exact value of cos 15 degrees.
Using the formula, cos 15 degrees = ±√[(1 + cos 30 degrees)/2].
Since we are dealing with a positive angle, we can take the positive square root to find the value of cos 15 degrees. Thus, cos 15 degrees is equal to √[(1 + cos 30 degrees)/2].
To find cos 30 degrees, we can refer to a trigonometric table, which states that cos 30 degrees is equal to √3/2. Plugging this value into our equation, cos 15 degrees becomes √[(1 + √3/2)/2].
Now that we have established the exact value of cos 15 degrees, we can return to our equation for sin 105 degrees.
sin 105 degrees = 1 * cos(15 degrees) + 0 * sin(15 degrees)
= 1 * √[(1 + √3/2)/2] + 0 * sin(15 degrees)
= √[(1 + √3)/2].
The exact value of sine 105 degrees is √[(1 + √3)/2].
FAQs:
1. How do you find the exact value of sine 105 degrees?
To find the exact value of sine 105 degrees, you can rewrite it as the sum of two known angles and apply trigonometric identities to simplify the expression. The result is √[(1 + √3)/2].
2. What is the half-angle formula?
The half-angle formula allows you to find the exact values of trigonometric functions for angles that are half the size of given angles.
3. Can I use a calculator to find the value of sine 105 degrees?
Yes, most calculators have a sine function that can provide an approximate value for sine 105 degrees. However, to find the exact value, the method outlined in this article must be used.
4. Why is it important to find the exact value?
Finding the exact value of trigonometric functions helps in various fields such as engineering, physics, and mathematics, where precise calculations are required.
5. How can I remember trigonometric identities?
Practicing and understanding the derivations of trigonometric identities can help you remember them. Additionally, creating flashcards or using mnemonic devices can be useful techniques.
6. Is it necessary to simplify the expression when finding trigonometric values?
Simplifying the expression allows you to present the answer in a concise and precise form. However, the calculation will still be valid without simplification.
7. Can the exact value of sine 105 degrees be expressed as a fraction?
Yes, the exact value of sine 105 degrees, √[(1 + √3)/2], can be expressed as a fraction to represent a seemingly irrational value.
8. How can trigonometric identities be applied to real-life situations?
Trigonometric identities find immense applications in physics, engineering, architecture, navigation, and computer graphics, enabling accurate calculations and measurements.
9. What other trigonometric functions can be found using this method?
The method showcased in this article can be employed to find the exact values of other trigonometric functions such as cosine, tangent, cosecant, secant, and cotangent.
10. Can the method be used to find the exact value of sine for any angle?
Yes, the method presented here is applicable to finding the exact value of sine for any given angle.
11. Is knowing trigonometric values helpful in solving triangles?
Yes, knowing trigonometric values assists in solving triangle problems involving angles and sides, as these functions relate the angles and sides of a triangle.
12. Are there alternative methods to find the exact value of sine 105 degrees?
There are alternative approaches, such as using power series expansions or evaluating the angle using complex numbers. However, the method described in this article is often the easiest and most straightforward solution.