{"id":218330,"date":"2025-03-09T06:50:14","date_gmt":"2025-03-09T06:50:14","guid":{"rendered":"https:\/\/namso-gen.co\/blog\/how-to-find-the-exact-value-of-sine-105-degrees\/"},"modified":"2025-03-09T06:50:14","modified_gmt":"2025-03-09T06:50:14","slug":"how-to-find-the-exact-value-of-sine-105-degrees","status":"publish","type":"post","link":"https:\/\/namso-gen.co\/blog\/how-to-find-the-exact-value-of-sine-105-degrees\/","title":{"rendered":"How to find the exact value of sine 105 degrees?"},"content":{"rendered":"

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.<\/p>\n

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).<\/p>\n

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).<\/p>\n

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).<\/p>\n

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).<\/p>\n

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.<\/p>\n

The half-angle formula states that cos(x\/2) = \u00b1\u221a[(1 + cos x)\/2]. By plugging in x = 30 degrees, we can determine the exact value of cos 15 degrees.<\/p>\n

Using the formula, cos 15 degrees = \u00b1\u221a[(1 + cos 30 degrees)\/2].<\/p>\n

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 \u221a[(1 + cos 30 degrees)\/2].<\/p>\n

To find cos 30 degrees, we can refer to a trigonometric table, which states that cos 30 degrees is equal to \u221a3\/2. Plugging this value into our equation, cos 15 degrees becomes \u221a[(1 + \u221a3\/2)\/2].<\/p>\n

Now that we have established the exact value of cos 15 degrees, we can return to our equation for sin 105 degrees.<\/p>\n

sin 105 degrees = 1 * cos(15 degrees) + 0 * sin(15 degrees)
\n= 1 * \u221a[(1 + \u221a3\/2)\/2] + 0 * sin(15 degrees)
\n= \u221a[(1 + \u221a3)\/2].<\/p>\n

The exact value of sine 105 degrees is \u221a[(1 + \u221a3)\/2].<\/b><\/p>\n

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