Question

Find the value of \[\sin (2{\sin ^{ - 1}}0.8)\]
A.0.48
B.\[\sin {1.2^ \circ }\]
C. \[\sin {1.6^ \circ }\]
D.0.96

Answer 1

Hint: First suppose that \[x = {\sin ^{ - 1}}0.8\] , then obtain the value of \[\sin x,{\rm{ cos}}x\] then apply the formula of \[\sin 2x\] and substitute the values of \[\sin x,{\rm{ cos}}x\] to obtain the required result.

Formula Used: \[{\sin ^2}x + {\cos ^2}x = 1\]
And \[\sin 2x = 2\sin x\cos x\]

Complete step by step solution: Suppose that \[x = {\sin ^{ - 1}}0.8\],so, \[\sin x = 0.8\] .
Hence,
\[\cos x = \sqrt {1 - {{\left( {0.8} \right)}^2}} \]
\[ = \sqrt {1 - 0.64} \]
\[ = \sqrt {0.36} \]
\[ = 0.6\]
Now, the given equation can be written as \[\sin 2x\].
So,
\[\sin 2x = 2\sin x\cos x\]
=\[2 \times \left( {0.8} \right) \times \left( {0.6} \right)\]
=\[0.96\]

Option ‘D’ is correct

Additional Information: Trigonometry ratios deal with the ratios the length of sides of a right angle triangle. There are six ratios that are sin, cos, sec, cosec, tan, cot. These trigonometry ratios are applicable for a right angle triangle. There are some identities in trigonometry. The identities are reciprocal trigonometric identities, Pythagorean trigonometric identities, ratio trigonometric identities, trigonometric identities of opposite angles, trigonometric identities of complementary angle, trigonometric identities of supplementary angle, sum and difference of angles trigonometric identities, double angle trigonometric identities, half angle trigonometric identities, product-sum trigonometric identities, trigonometric identities of product.

Note: Sometime students directly substitute the value of \[{\sin ^{ - 1}}0.8\]by the help of a calculator then multiply that by 2 and obtain the sine value of that function, in exam we are not allowed to use calculators so do the calculation as shown.