How to find the exact value of sine 11pi/12?

Trigonometry is an essential branch of mathematics that focuses on the relationships and properties of triangles. Sine, one of the fundamental trigonometric functions, helps us determine the relationship between angles and the ratios of the lengths of the sides of a right triangle.

However, in some cases, we encounter angles that are not part of the typical unit circle angles, such as 11π/12. Finding the exact value of the sine of these angles may seem challenging at first, but with a systematic approach, it becomes much more manageable.

Understanding the Problem

To find the exact value of sine 11π/12, we must break down the angle into simpler terms and utilize trigonometric identities and the unit circle.

Let’s break 11π/12 into two angles: 3π/4 and π/6, which are angles we can easily work with using the unit circle.

Using the Angle-Sum Identity

The angle-sum identity for sine states that sin(A + B) = sin(A)cos(B) + cos(A)sin(B). We can exploit this identity to our advantage by substituting known angles into this equation and solving it step by step.

How do we apply the angle-sum identity?

To apply the angle-sum identity, we use known angles and their corresponding sine and cosine values.

Using the angle-sum identity sin(A + B) = sin(A)cos(B) + cos(A)sin(B) for our case, where A = 3π/4 and B = π/6, we can derive the expression for sin(11π/12).

What are the sine and cosine values of 3π/4 and π/6?

The sine and cosine values of 3π/4 are: sin(3π/4) = 1/√2 and cos(3π/4) = -1/√2.
The sine and cosine values of π/6 are: sin(π/6) = 1/2 and cos(π/6) = √3/2.

Breaking Down sin(11π/12)

Now, let’s substitute the corresponding sine and cosine values into our expression.

sin(11π/12) = sin(3π/4 + π/6) = sin(3π/4)cos(π/6) + cos(3π/4)sin(π/6)

Substituting the values, we get:

sin(11π/12) = (1/√2)(√3/2) + (-1/√2)(1/2)
= √3/2√2 – 1/2√2
= (√3 – 1)/(2√2)

How do we simplify (√3 – 1)/(2√2) further?

To simplify (√3 – 1)/(2√2), we multiply both the numerator and denominator by √2 to eliminate the radical from the denominator.

(√3 – 1)/(2√2) = (√(3) – 1)(√(2))/(2√(2))(√(2))
= (√(6) – √(2))/(2√(4))
= (√(6) – √(2))/(4)

The Exact Value of sine 11π/12 is (√6 – √2)/4.

We have successfully determined the exact value of sine 11π/12, which is (√6 – √2)/4.

Now that we have found the answer to the primary question, let’s address some related frequently asked questions:

Related FAQs:

1. How can I find the sine of any angle?

The sine of an angle can be found by dividing the length of the side opposite the angle by the length of the hypotenuse in a right triangle.

2. Where can I find a unit circle?

Numerous resources, including textbooks and online references, provide diagrams and explanations of the unit circle.

3. What is the unit circle?

The unit circle is a circle drawn on a coordinate plane, centered at the origin (0,0), with a radius of 1 unit.

4. How can I determine the value of sine for angles larger than 2π?

You can convert angles larger than 2π into their equivalent angles between 0 and 2π using the modulo operation (angle % 2π).

5. Are there other trigonometric identities that can be used?

Yes, there are several identities such as the angle-difference identity, double-angle identity, and Pythagorean identities that can be utilized depending on the given problem.

6. Can I use a calculator to find the exact value of sine 11π/12?

While a calculator can provide an approximate value for sine 11π/12, finding the exact value relies on trigonometric identities and the unit circle.

7. How can I remember the trigonometric ratios?

Reviewing and practicing trigonometric ratios and their properties through various exercises and problems can help you remember them.

8. What other trigonometric functions are commonly used?

Apart from sine, cosine, and tangent, other trigonometric functions include cosecant, secant, and cotangent.

9. How are trigonometric functions used in real-life applications?

Trigonometric functions have numerous applications in physics, engineering, construction, and navigation, among others.

10. Is it possible to find the exact value of sine π/3?

Yes, the exact value of sine π/3 is √3/2.

11. How are the values of sine and cosine related in the unit circle?

The values of sine and cosine are related as complementary angles in the unit circle. The sine of an angle is equal to the cosine of its complementary angle.

12. Can trigonometry be used in non-right triangles?

Yes, the branch of trigonometry called “trigonometry of general triangles” deals with non-right triangles and defines further trigonometric functions such as the law of sines and the law of cosines.

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