Question

What does sin cos and tan mean?

Answer 1

`\text{sine(sin), cosine(cos), tangent(tan), cotangent(cot), secant(sec) and }\\\text{cosecant(cosec) are the trigonometric ratios/functions that relate the angles}\\\text{and sides of a right triangle.}`

`\text{Based on the reference angle taken in a right triangle, the perpendicular varies}\\\text{which consequently affect these trigonometric ratios. See the image for more }\\\text{detail.}`

`\text{The right angle which is opposite to the hypotenuse cannot be taken as a}\\\text{reference.}`

`\text{Here are the values of all the trigonometric ratios in a right triangle. }\\\text{sin}\theta=\text{p/h}\\\text{cos}\theta=\text{b/h}\\\text{tan}\theta=\text{p/b}\\\text{cot}\theta=\text{b/p}\\}\text{sec}\theta=\text{h/b}\\}\text{cosec}\theta=\text{h/p}`

`\text{Here, }\theta\text{ is the angle taken as a reference in the right triangle. The opposite}\\\text{side of the reference angle is perpendicular and the other leg is taken as base.}`

`\text{Here are some important formula we may derive from these ratios:}\\\text{cosec}\theta=\text{h/p}=\frac{1}{\text{p/h}}=\text{1/sin}\theta\\\\\text{sec}\theta=\mathrm{h/b = (1)/(b/h)=1/cos}\theta\\\\\text{cot}\theta=\mathrm{b/p=(1)/(p/b)=1/tan}\theta\\\\\text{sin}\theta/\text{cos}\theta=\mathrm{(p/h)/(b/h)=p/b=tan\theta}\\\\\mathrm{cos\theta/sin\theta=(b/h)/(p/h)=b/p=cot\theta}`

`\text{Furthermore, from the pythagoras theorem we have}\\\mathrm{h^2=p^2+b^2}.......(1)\\\text{Dividing equation(1) by h}^2,\\\mathrm{(h^2)/(h^2)=(p^2)/(h^2)+(b^2)/(h^2)}\\\\\mathrm{or,\ 1=((p)/(h))^2+((b)/(h))^2}\\\\\mathrm{or,\ sin^2\theta+cos^2\theta=1}....(2)`

`\text{Dividing equation(2) by sin}^2\theta,\\\mathrm{sin^2\theta/sin^2\theta+cos^2\theta/sin^2\theta=1/sin^2\theta}\\\mathrm{or,\ 1+cot^2\theta=cosec^2\theta}\\\therefore\ \mathrm{cosec^2\theta-cot^2\theta=1......(3)}`

`\text{Dividing equation(2) by cos}^2\theta,\\\mathrm{sin^2\theta/cos^2\theta+cos^2\theta/cos^2\theta=1/cos^2\theta}\\\mathrm{or,\ tan^2\theta+1=sec^2\theta}\\\mathrm{or,\ sec^2\theta-tan^2\theta=1}`

`\text{Note that (sin}\theta)^n=\mathrm{sin^n\theta\ unless\ n=-1.}\text{ This rule applies to every other}\\\text{trigonometric ratios whether cos, tan, sec or cosec. Because for example, }\\\text{(cos}\theta)^(-1)\text{ simply means }\text{sec}\theta\text{ whereas cos}^(-1)\theta\text{ is an inverse function. You will}\\\text{learn about the inverse trigonometric functions later in higher grades.}`

Answer 2

Sine, cosine, and tangent are the three primary trigonometric functions used to relate the angles and sides of a right triangle. They are abbreviated as sin, cos, and tan, respectively.

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Here is a brief explanation of each function:

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Sine: The sine of an angle is the ratio of the length of the side opposite the angle to the length of the hypotenuse of the triangle.

Cosine: The cosine of an angle is the ratio of the length of the side adjacent to the angle to the length of the hypotenuse of the triangle.

Tangent: The tangent of an angle is the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle.

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These functions are used in trigonometry to solve problems related to triangles, such as finding the length of a side or the measure of an angle. They are also used in calculus when dealing with graphs of waves .

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