Proof. The area of any triangle can be written as one half of its base times its height. Selecting one side of the triangle as the base, the height of the triangle relative to that base is computed as the length of another side times the sine of the angle between the chosen side and the base.
Law of Sines Proof
Given: △ABC, AB = c, BC = a and AC = b. Construction: Draw a perpendicular, CD ⊥ AB. Then CD = h is the height of the triangle. “h” separates the △ ABC in two right-angled triangles, △CDA and △CDB. To Show: a / b = Sin A / Sin B.
To prove the Sine Rule, consider three identical copies of the same triangle with sides a,b,c and (opposite) angles A,B,C. Divide each into two right angled triangles. To prove the Cosine Rule, consider three identical copies of the same triangle with sides a,b,c and (opposite) angles A,B,C.
Definition of the Law of Sines: If A, B, and C are the measurements of the angles of an oblique triangle, and a, b, and c are the lengths of the sides opposite of the corresponding angles, then the ratios of the a side's length to the sine of the angle opposite the side must all be the same.
In trigonometry, the name “sine” comes through Latin from a Sanskrit word meaning “chord”. In the picture of a unit circle below, AB has length sinθ and this is half a chord of the circle.
To solve a triangle is to find the lengths of each of its sides and all its angles. The sine rule is used when we are given either a) two angles and one side, or b) two sides and a non-included angle. The cosine rule is used when we are given either a) three sides or b) two sides and the included angle.
The Law of Sines can be used to solve oblique triangles, which are non-right triangles. According to the Law of Sines, the ratio of the measurement of one of the angles to the length of its opposite side equals the other two ratios of angle measure to opposite side.
Using trigonometry
This proof uses trigonometry in that it treats the cosines of the various angles as quantities in their own right. It uses the fact that the cosine of an angle expresses the relation between the two sides enclosing that angle in any right triangle.
The sine rule
We are now going to extend trigonometry beyond right angled triangles and use it to solve problems involving any triangle. We can use the sine rule when we're given the sizes of: two sides and one angle (which is opposite to one of these sides) one side and any two angles.
Let us start with some definitions. We will call the ratio of the opposite side of a right triangle to the hypotenuse the sine and give it the symbol sin. sin = o / h. The ratio of the adjacent side of a right triangle to the hypotenuse is called the cosine and given the symbol cos.
Answer and Explanation: In order to calculate the trigonometric ratios of an acute angle of a triangle, the triangle must have a hypotenuse, and only right triangles have a hypotenuse. Therefore, we can only calculate trigonometric ratios of an acute angle using right triangles.
If side length 𝑎 is equal to the height ℎ or side length 𝑎 is greater than side length 𝑏, then one triangle can be formed. Finally, if the height of the triangle ℎ is less than side length 𝑎 which is less than side length 𝑏, then two triangles can be formed.
The sine of an angle is the trigonometric ratio of the opposite side to the hypotenuse of a right triangle containing that angle. sine=length of the opposite to the anglelength of the hypotenuse abbreviated as “sin”. Example: In the triangle shown, sinA=610 or 35 and sinB=810 or 45 .
As we learned, sine is one of the main trigonometric functions and is defined as the ratio of the side of the angle opposite the angle divided by the hypotenuse. It's important for finding distances or height and can also be used to find angle measures, which are measured in radians.
As sine of the angle 90° is 1, it is equal to the function sin 1. So, the inverse function of sin 1 is denoted as 90° or π/2. It is the highest value of the sine function.
The more general definition of the function is that the sine of an angle is equal to the height of the point on the unit circle given the correct angle, and since the unit circle reaches only heights between −1 and 1, so does the sine function.
You can use the sine rule to find the length of a side when its opposite angle and another opposite side and angle are given. To calculate an unknown side use the formula . Alternatively, you can use the sine rule to find an unknown angle if the opposite side and another opposite side and angle are given.
Sine is "opposite over hypotenuse" (the SOH of SOHCAHTOA). When we draw the triangle inside a unit circle the hypotenuse is automatically 1 at any angle. That means the sine of an angle is simply the length of the "opposite" leg of the triangle (opposite / 1).
The first trigonometric table was apparently compiled by Hipparchus of Nicaea (180 – 125 BCE), who is now consequently known as "the father of trigonometry." Hipparchus was the first to tabulate the corresponding values of arc and chord for a series of angles.
The shape of the graph turns out to be exactly the same as the sine function graph, the only difference in the two being that the cosine graph starts out at y = 1 for x = 0 while the sine graph started out at y = 0 for x = 0.