sinx cosx tanx

Answer

Sinx, Cosx, and Tanx are fundamental trigonometric functions used in mathematics.

• Sinx represents the ratio of the Opposite side to the Hypotenuse side in a right-angled triangle.


• Cosx represents the ratio of the Adjacent side to the Hypotenuse side in a right-angled triangle.


• Tanx represents the ratio of the Opposite side to the Adjacent side in a right-angled triangle.

These functions are extensively used in trigonometric identities, calculus, and geometry.

Full explanation

Sinx, Cosx, and Tanx are the three primary trigonometric ratios used in mathematics. These functions revolve around the concepts of right-angled triangles. To understand these functions, these triangle names will be helpful:



Here, the two perpendicular sides are called adjacent and opposite sides, and the longest side opposite to the right angle is called the hypotenuse.
Therefore, using this triangle, let's derive the primary trigonometric functions.

1. Sinx
Sinx represents the ratio of the Opposite side to the Hypotenuse side of a given right-angled triangle. Mathematically,
sinx = Opposite side / Hypotenuse side
For instance, if a right-angled triangle has an opposite side of 5 and a hypotenuse side of 10,
then sinx = 5/10, which simplifies to 1/2.
So, the value of sinx is 1/2 (or 0.5) for this triangle.

2. Cosx
Cosx represents the ratio of the Adjacent side to the Hypotenuse side in a right-angled triangle. Mathematically,
cosx = Adjacent side / Hypotenuse side
For example, in a right-angled triangle with an adjacent side of 8 and a hypotenuse side of 10,
cosx = 8/10, which simplifies to 4/5.
Thus, the value of cosx is 4/5 (or 0.8) for this triangle.

3. Tanx
Tanx represents the ratio of the Opposite side to the Adjacent side in a right-angled triangle. Mathematically,
tanx = Opposite side / Adjacent side
For instance, if a right-angled triangle has an adjacent side of 12 and an opposite side of 5,
tanx = 5/12, which simplifies to almost 0.42.
Thus, the value of tanx is 0.42 (rounded to the nearest hundredth) for this triangle.

All these trigonometric ratios or functions, i.e., sinx, cosx, and tanx, can be calculated for different angles. For example, sin60, cos45, and tan30 are some numeric values of these trigonometric functions that can be calculated for those specific angles. Also, these functions lead to trigonometric identities, allowing for more advanced applications in calculus, linear algebra, and more.

FAQs

1. What is the cosecant function of an angle x?
The Cosecant function of an angle x (cscx) represents the ratio of the Hypotenuse side to the Opposite side in a given right-angled triangle. In mathematical terms,
cscx = Hypotenuse side / Opposite side

2. How to calculate the value of sin2x?
To calculate sin2x, we can use the double-angle trigonometric identity for sine, which is
sin(2x) = 2sinxcosx.

3. Why is the domain of tanx limited?
The domain of tanx is limited to exclude odd multiples of pi/2 since at those points, the adjacent side is zero, and the ratio becomes undefined.

Conclusion

Sinx, Cosx, and Tanx are the primary trigonometric functions that help understand ratios in right-angled triangles. They have applications in various fields like engineering, physics, and computer graphics. Being able to understand and apply trigonometric functions is essential in solving problems involving space, distance, and angles. Therefore, learning and mastering these functions is incredibly beneficial to anyone studying mathematics.