What Is $\sin 28^{\circ}$?A. $\frac{15}{8}$ B. $\frac{8}{15}$ C. $\frac{8}{17}$ D. $\frac{15}{17}$

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Understanding Trigonometric Functions: A Comprehensive Guide to Calculating sin⁡28∘\sin 28^{\circ}

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. It is a fundamental subject that has numerous applications in various fields, including physics, engineering, and navigation. One of the most important concepts in trigonometry is the sine function, which is used to calculate the ratio of the length of the side opposite an angle to the length of the hypotenuse in a right-angled triangle. In this article, we will explore the concept of the sine function and use it to calculate sin⁡28∘\sin 28^{\circ}.

The sine function is a mathematical function that is used to calculate the ratio of the length of the side opposite an angle to the length of the hypotenuse in a right-angled triangle. It is denoted by the symbol sin⁡\sin and is defined as:

sin⁡θ=oppositehypotenuse\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}

where θ\theta is the angle being measured, and "opposite" and "hypotenuse" refer to the lengths of the sides of the triangle.

To calculate sin⁡28∘\sin 28^{\circ}, we need to use a trigonometric table or a calculator that can perform trigonometric calculations. A trigonometric table is a table that lists the values of the sine, cosine, and tangent functions for various angles. We can use this table to find the value of sin⁡28∘\sin 28^{\circ}.

Alternatively, we can use a calculator that has a trigonometric function button. Most scientific calculators have a button that allows us to enter an angle and calculate the sine, cosine, or tangent of that angle.

Using a Trigonometric Table

A trigonometric table is a table that lists the values of the sine, cosine, and tangent functions for various angles. We can use this table to find the value of sin⁡28∘\sin 28^{\circ}. Here is a sample trigonometric table:

Angle sin⁡θ\sin \theta cos⁡θ\cos \theta tan⁡θ\tan \theta
0∘0^{\circ} 0 1 0
10∘10^{\circ} 0.1736 0.9848 0.1763
20∘20^{\circ} 0.3420 0.9397 0.3640
30∘30^{\circ} 0.5000 0.8660 0.5774
40∘40^{\circ} 0.6428 0.7660 0.8367
50∘50^{\circ} 0.7660 0.6428 1.1945
60∘60^{\circ} 0.8660 0.5000 1.7321
70∘70^{\circ} 0.9397 0.3420 2.7445
80∘80^{\circ} 0.9848 0.1736 5.7071
90∘90^{\circ} 1 0 undefined

We can see from the table that the value of sin⁡28∘\sin 28^{\circ} is approximately 0.4695.

Using a Calculator

Alternatively, we can use a calculator that has a trigonometric function button. Most scientific calculators have a button that allows us to enter an angle and calculate the sine, cosine, or tangent of that angle.

To calculate sin⁡28∘\sin 28^{\circ} using a calculator, we need to follow these steps:

  1. Enter the angle 28∘28^{\circ} into the calculator.
  2. Press the "sin" button to calculate the sine of the angle.
  3. The calculator will display the value of sin⁡28∘\sin 28^{\circ}.

In conclusion, the sine function is a mathematical function that is used to calculate the ratio of the length of the side opposite an angle to the length of the hypotenuse in a right-angled triangle. We can use a trigonometric table or a calculator to calculate the value of sin⁡28∘\sin 28^{\circ}. The value of sin⁡28∘\sin 28^{\circ} is approximately 0.4695.

The correct answer is:

1517\boxed{\frac{15}{17}}

This is the value of sin⁡28∘\sin 28^{\circ}, which is approximately 0.4695.
Frequently Asked Questions: Understanding Trigonometric Functions

A: The sine function is a mathematical function that is used to calculate the ratio of the length of the side opposite an angle to the length of the hypotenuse in a right-angled triangle. It is denoted by the symbol sin⁡\sin and is defined as:

sin⁡θ=oppositehypotenuse\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}

A: There are two ways to calculate the sine of an angle: using a trigonometric table or using a calculator. A trigonometric table is a table that lists the values of the sine, cosine, and tangent functions for various angles. We can use this table to find the value of the sine of an angle. Alternatively, we can use a calculator that has a trigonometric function button. Most scientific calculators have a button that allows us to enter an angle and calculate the sine, cosine, or tangent of that angle.

A: The sine and cosine functions are two trigonometric functions that are used to calculate the ratios of the sides of a right-angled triangle. The sine function is used to calculate the ratio of the length of the side opposite an angle to the length of the hypotenuse, while the cosine function is used to calculate the ratio of the length of the side adjacent to an angle to the length of the hypotenuse.

A: To use a trigonometric table to calculate the sine of an angle, follow these steps:

  1. Find the angle in the table that is closest to the angle you want to calculate.
  2. Look up the value of the sine function for that angle in the table.
  3. Use the value from the table as an approximation of the sine of the angle you want to calculate.

A: To use a calculator to calculate the sine of an angle, follow these steps:

  1. Enter the angle you want to calculate into the calculator.
  2. Press the "sin" button to calculate the sine of the angle.
  3. The calculator will display the value of the sine of the angle.

A: The range of the sine function is the set of all possible values that the sine function can take. The range of the sine function is the interval [−1,1][-1, 1].

A: The domain of the sine function is the set of all possible angles for which the sine function is defined. The domain of the sine function is the set of all real numbers.

A: To graph the sine function, follow these steps:

  1. Plot the point (0,0)(0, 0) on the graph.
  2. Plot the point (π2,1)(\frac{\pi}{2}, 1) on the graph.
  3. Plot the point (π,0)(\pi, 0) on the graph.
  4. Plot the point (3π2,−1)(\frac{3\pi}{2}, -1) on the graph.
  5. Connect the points with a smooth curve to form the graph of the sine function.

A: The inverse sine function is a function that takes a value in the range of the sine function and returns the angle for which the sine function takes that value. The inverse sine function is denoted by the symbol sin⁡−1\sin^{-1} and is defined as:

sin⁡−1x=θ if and only if sin⁡θ=x\sin^{-1} x = \theta \text{ if and only if } \sin \theta = x

A: To calculate the inverse sine of a value, follow these steps:

  1. Enter the value you want to calculate the inverse sine of into the calculator.
  2. Press the "sin^-1" button to calculate the inverse sine of the value.
  3. The calculator will display the value of the inverse sine of the value.

A: The inverse sine and sine functions are two trigonometric functions that are used to calculate the ratios of the sides of a right-angled triangle. The sine function is used to calculate the ratio of the length of the side opposite an angle to the length of the hypotenuse, while the inverse sine function is used to calculate the angle for which the sine function takes a given value.