In trigonometry, sin, cos, và tan are the basic trigonometric ratios used khổng lồ study the relationship between the angles và sides of a triangle (especially of a right-angled triangle). Pythagoras worked on the relationship between the sides of a right triangle through the Pythagorean theorem while Hipparcus worked on establishing the relationship between the sides và angles of a right triangle using the concepts of trigonometry. Sin, cos, và tan formulas in trigonometry are used to lớn find the missing sides or angles of a right-angled triangle.
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Let's understand the sin, cos, và tan in trigonometry using formulas & examples.
|1.||What is Sin Cos rã in Trigonometry?|
|2.||Sin Cos rã Formula|
|3.||Sin Cos tung Table|
|4.||Tips to Remember Sin Cos tan Table|
|5.||Sin Cos rã on Unit Circle|
|6.||Application of Sin Cos tan in Real Life|
|7.||FAQs on sin cos tan|
What is Sin Cos tan in Trigonometry?
Sin, cos, và tan are the three primary trigonometric ratios, namely, sine, cosine, and tangent respectively, where each of which gives the ratio of two sides of a right-angled triangle. We know that the longest side of a right-angled triangle is known as the "hypotenuse" and the other two sides are known as the "legs." That means, in trigonometry, the longest side of a right-angled triangle is still known as the "hypotenuse" but the other two legs are named to lớn be:opposite side andadjacent side
We decide the "opposite" và "adjacent" sides based upon the angle which we are talking about.The "opposite side" or the perpendicular is the side that is just "opposite" to the angle.The "adjacent side" or the base is the side(other than the hypotenuse) that "touches" the angle.
Sin Cos tan Values
Sin, Cos, và Tan values in trigonometry refer khổng lồ the values of the respective trigonometric function for the given angle. We can find the sin, cos và tan values for a given right triangle by finding the required ratio of the sides. Let us understand the formulas to find these ratios in detail in the following sections.
Sin Cos tung Formulas
Sin, cos, và tan functions in trigonometry are defined in terms of two of the three sides (opposite, adjacent, & hypotenuse) of a right-angled triangle. Here are the formulas of sin, cos, và tan.
sin θ = Opposite/Hypotenuse
cos θ = Adjacent/Hypotenuse
tan θ = Opposite/Adjacent
Apart from these three trigonometric ratios, we have another three ratios called csc, sec, and cot which are the reciprocals of sin, cos, & tan respectively. Let us understand these sin, cos, và tan formulas using the example given below.
Example: Find the sin, cos, and tan of the triangle for the given angle θ.
In the triangle, the longest side (or) the side opposite lớn the right angle is the hypotenuse. The side opposite lớn θ is the opposite side or perpendicular. The side adjacent khổng lồ θ is the adjacent side or base.
Now we find sin θ, cos θ, & tan θ using the above formulas:
sin θ = Opposite/Hypotenuse = 3/5
cos θ = Adjancent/Hypotenuse = 4/5
tan θ = Opposite/Adjacent = 3/4
Trick khổng lồ remember sin cos tung formulas in trigonometry: Here is a trick lớn remember the formulas of sin, cos, và tan. We can use the acronym "SOHCAHTOA" as shown below,
Sin Cos tung Table
The trigonometric ratios, sin, cos, & tan do not exactly depend upon the side lengths of the triangle but rather they depend upon the angle because ultimately, we are taking the ratio of the sides. Sin, cos, & tan table is used khổng lồ find the value of these trigonometric functions for the standard angles. During calculations involving sine, cosine, or tangent ratios, we can directly refer to the trig chart given in the following section to make the deductions easier.
Sin Cos tan Chart
Sin cos tan chart/table is a chart with the trigonometric values of sine, cosine, và tangent functions for some standard angles 0o, 30o, 45o, 60o, & 90o. We can refer to lớn the trig table given below to directly pick values of sin, cos, và tan values for standard angles.
Tips to Remember Sin Cos tan Table
The tips that you need to lớn memorize from this chart are:The angles 0o, 30o, 45o, 60o, and 90o in order.The first row (of sin) can be remembered like this: 0/2, √1/2, √2/2, √3/2.That's all you need to remember because:The row of cos is as same as the row of sin just in the reverse order.Each value in the row of tan is obtained by dividing the corresponding values of sin by cos because rã = sin/cos.
You can see how is chảy = sin/cos here:
sin θ/cos θ = (Opposite/Hypotenuse) ÷ (Adjacent/Hypotenuse) = (Opposite/Hypotenuse) × (Hypotenuse/Adjacent) = Opposite/Adjacent = chảy θ
Sin Cos tan on Unit Circle
The values of sin, cos, & tan can be calculated for any given angle using the unit circle. Unit circle in a coordinate plane is a circle of unit radius of 1, frequently centered at the origin (0, 0) in the Cartesian coordinate system in the Euclidean plane, especially in trigonometry. For any point on unit circle, given with the coordinates(x, y), the sin, cos và tan ratios can be given as,sin θ = y/1cos θ = x/1tan θ = y/x
where θ is the angle the line joining the point and origin forms with the positive x-axis.
Application of Sin Cos chảy in Real Life
Trigonometry ratios sin, cos, rã find application in finding heights and distances in our daily lives. We use sin, cos, and tan to solve many real-life problems. Here is an example to lớn understand the applications of sin, cos và tan.
Example: A ladder leans against a brick wall making an angle of 50o with the horizontal. If the ladder is at a distance of 10 ft from the wall, then up to lớn what height of the wall the ladder reaches?
Let us assume that the ladder reaches till x ft of the wall.
Using the given information:
Here, we know the adjacent side (which is 10 ft) & we have khổng lồ find the opposite side (which is x ft). So we use the relation between the opposite and the adjacent sides which is tan.
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tan 50o = x/10
x = 10 tung 50o
x ≈ 11.9 ft
Here, tung 50o is calculated using the calculator and the final answer is rounded up khổng lồ 1 decimal. Therefore, the ladder reaches up to lớn 11.9 ft of the wall.