__maths__>__Trigonometry__>__Trigonometric Ratios for Standard Angles__### Trigonometric Ratios for Standard Angles

Students need not memorize a table of trigonometric ratios for standard angles and instead they can quickly calculate the ratios for standard angle. This page explains how to quickly calculate.

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Given the right angled triangle `Delta OPQ`, and `/_POQ = 45^@`.

`bar(OP)= 1`

`bar(OQ)=bar(PQ)=x=y`

`bar(OQ)^2+bar(PQ)^2 = bar(OP)^2 = 1`

`2 x^2 = 1`

The point P is `(1/sqrt(2), 1/sqrt(2))`.

It is proven that point `P (1/sqrt(2), 1/sqrt(2))`. That is,

opposite side `bar(PQ) = 1/sqrt(2)`

adjacent side `bar(OQ) = 1/sqrt(2)`

hypotenuse `bar(OP) = 1` `sin 45^@ = 1/sqrt(2) `

`cos 45^@ = 1/sqrt(2) `

`tan 45^@ = 1`

Do not memorize the trigonometric ratios, it is easy to work out the lengths of the three sides of a triangle of standard angles.

Once we know the lengths of the three sides, it is easy to work out the trigonometric ratios.

For example, `cot 45^@` is `1` (by finding ratio adjacent by opposite side).

Given the right angled triangle `Delta OPQ`, and `/_POQ = 30^@`.

`bar(OP)= 1`

Construct an equilateral triangle `Delta OP′P`.

From this, it is derived that `bar(PQ)=1/2`.

`bar(OQ)^2+bar(PQ)^2 = bar(OP)^2 = 1`

`x^2+(1/2)^2 = 1`

The point `P` is `(sqrt(3)/2 , 1/2)`.

It is proven that point `P (sqrt(3)/2 , 1/2)`. That is,

opposite side `bar(PQ) = 1/2`

adjacent side `bar(OQ) = sqrt(3)/2`

hypotenuse `bar(OP) = 1` `sin 30^@ = 1/2`

`cos 30^@ = sqrt(3)/2`

`tan 30^@ = 1/sqrt(3)`

It is noted that one does not need to memorize the trigonometric ratios, it is easy to work out the lengths of the three sides of a triangle of standard angles. Once we know the lengths of the three sides, it is easy to work out the trigonometric ratios.

For example, `sec 30^@ = 2/sqrt(3)`

Given the right angled triangle `Delta OPQ`, and `/_POQ = 60^@`.

`bar(OP)= 1`

Construct an equilateral triangle `Delta O Q′P`.

From this, it is derived that `bar(OQ)=1/2`.

`bar(OQ)^2+bar(PQ)^2 = bar(OP)^2 = 1`

`(1/2)^2+ y^2 = 1`

The point `P` is `(1/2 , sqrt(3)/2)`.

It is proven that point `P (1/2 , sqrt(3)/2)`. That is,

opposite side `bar(PQ) = sqrt(3)/2`

adjacent side `bar(OQ) = 1/2`

hypotenuse `bar(OP) = 1` `sin 60^@ = sqrt(3)/2`

`cos 60^@ = 1/2`

`tan 60^@ = sqrt(3)`

It is repeated that one does not need to memorize the trigonometric ratios, it is easy to work out the lengths of the three sides of a triangle of standard angles and thus the ratios.

For example, the `text(cosec) 60^@ = 2/sqrt(3)`

Given the right angled triangle `Delta OPQ`, and `/_POQ = 0^@`.

`bar(OQ)=bar(OP)= 1`

The point `P` is `(1,0)`

It is proven that point `P (1, 0)`. That is,

opposite side `bar(PQ) = 0`

adjacent side `bar(OQ) = 1`

hypotenuse `bar(OP) = 1` `sin 0^@ = 0`

`cos 0^@ = 1`

`tan 0^@ = 0`

It might not be intuitive, but it is possible to work out the lengths of the three sides of a triangle of `0^@` degree and work out the trigonometric ratios without memorizing them.

For example, `sec 0^@ = 1`.

Given the right angled triangle `Delta OPQ`, and `/_POQ = 90^@`.

`bar(OP) = bar(QP) = 1`

The point `P` is `(0,1)`.

It is proven that point `P (0,1)`. That is,

opposite side `bar(PQ) = 1`

adjacent side `bar(OQ) = 0`

hypotenuse `bar(OP) = 1` `sin 90^@ = 1`

`cos 90^@ = 0`

`tan 90^@ = 1/0 `

Another triangle that is not very intuitive, but it is possible to work out the lengths of the three sides of a triangle of `90^@` and work out the trigonometric ratios without memorizing them.

For example, `text(cosec) 90^@ = 1`.

Trigonometric ratios for all standard angles is captured in the figure. When a trigonometric ratio is required for an angle, quickly work out the `x,y` for the angle. And sine is chord, cosine is the chord on the complementary angle, tan is the tangent etc. With a little bit of practice, this becomes fast and easy. I hope you do not memorize a table which you will eventually forget. If you can work out from the first principles, the information will last a lot longer. With a little bit of practice, students can recall the ratios as fast as when memorized.

To compute trigonometric ratios for standard angles, use the properties of the triangles having the given angle.

**Trigonometric Ratios of Standard Angles: **

When a trigonometric ratio is required for an angle, quickly work out the `x,y` for the angle.

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