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| Trigonometrical Ratios of Standard Angles |
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| 0o, 30o, 45o, 60o and 90o are called standard angles. |
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| These angles are called standard angles because it is possible to obtain simple mathematical ratios for these angles. |
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| The students are advised to learn the T-ratios of these angles, they can also obtain them as shown in the derivation. |
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Let  |
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| \ AB = BC = a |
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| Using Pythagoras Theorem |
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| AC2 = a2 + a2 = 2a2 |
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Let  and  |
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| In 30o - 60o - 90o triangle, it can be proved that the hypotenuse is double the side opposite to 30o (see proof in Geometry Section), |
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| AC = 2AB |
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| Let AB = a |
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 AC = 2a |
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| Using Pythagoras Theorem |
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| BC2 = (2a)2 - a2 |
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| = 4a2 - a2 |
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| = 3a2 |
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| Draw a circle with radius r and XOX' and YOY' as axes. |
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Let  |
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Let  |
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| We observe A"B" > A'B' > AB, denominator remains 'r' (radius). |
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 As the angle increases sine ratio increases. |
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| We observe OB'' < OB' < OB, denominator remains r. |
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| As the angle increases cosine ratio decreases, |
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| When a = 0, |
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When  |
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 (infinity) |
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| As the angle increases tangent ratio increases. |
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| From the above results we conclude that for an acute angle q, the following results hold good. |
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(i)  |
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(ii)  |
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(iii)  |
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| Aid to memorise the table for standard angles: |
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| (a) Write 0, 1, 2, 3, 4 over each column as shown above. |
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| (b) Divide each number by 4 and take the square root. The values obtained are sine ratios. |
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| (c) Write these ratios in reverse order and you obtain the cosine ratios |
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| (sin A = cos B if A + B = 90o) |
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| (d) Divide each sine ratio by cosine ratio and you obtain the values of corresponding tangent ratios |
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Without using tables, find the value of  |
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| If A = 30o, verify sin 2A = 2 sin A cos A. |
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| L.H.S sin 2A = sin 2 x 30 |
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| = sin 60 |
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| R.H.S 2 sin A cos A = 2 sin 30 cos 30 |
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 sin 2A = 2 sin A cos A |
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| Hence L.H.S. = R.H.S. |
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