Trigonometrical Ratios of Standard Angles

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T-ratios of standard angles

0^o, 30^o, 45^o, 60^o and 90^o are called standard angles.

These angles are called standard angles because it is possible to obtain simple mathematical ratios for these angles.

The students are advised to learn the T-ratios of these angles, they can also obtain them as shown in the derivation.

T-ratios of 45^o

Let

\ AB = BC = a

Using Pythagoras Theorem

AC² = a² + a² = 2a²

T-ratios of 30^o and 60^o

Let and

In 30^o - 60^o - 90^o triangle, it can be proved that the hypotenuse is double the side opposite to 30^o (see proof in Geometry Section),

AC = 2AB

Let AB = a

AC = 2a

Using Pythagoras Theorem

BC² = (2a)² - a²

= 4a² - a²

= 3a²

T-ratios of 0^o and 90^o

Draw a circle with radius r and XOX^' and YOY^' as axes.

Let

Let

We observe A"B" > A'B' > AB, denominator remains 'r' (radius).

As the angle increases sine ratio increases.

We observe OB'' < OB' < OB, denominator remains r.

As the angle increases cosine ratio decreases,

When a = 0,

When

(infinity)

As the angle increases tangent ratio increases.

From the above results we conclude that for an acute angle q, the following results hold good.

(i)

(ii)

(iii)

Aid to Memory

Aid to memorise the table for standard angles:

(a) Write 0, 1, 2, 3, 4 over each column as shown above.

(b) Divide each number by 4 and take the square root. The values obtained are sine ratios.

(c) Write these ratios in reverse order and you obtain the cosine ratios

(sin A = cos B if A + B = 90^o)

(d) Divide each sine ratio by cosine ratio and you obtain the values of corresponding tangent ratios

Without using tables, find the value of

If A = 30^o, verify sin 2A = 2 sin A cos A.

L.H.S sin 2A = sin 2 x 30

= sin 60

R.H.S 2 sin A cos A = 2 sin 30 cos 30

sin 2A = 2 sin A cos A

Hence L.H.S. = R.H.S.