9. 2sin(t) + 3cos(t) = (P - 3)/2
Understand the Problem
The question involves solving a mathematical expression involving trigonometric functions, likely requiring simplification or evaluation of the expression provided.
Answer
The simplified expression is \( \sqrt{13} \sin\left(t + \tan^{-1}\left(\frac{3}{2}\right)\right) \).
Answer for screen readers
The simplified expression is ( \sqrt{13} \sin\left(t + \tan^{-1}\left(\frac{3}{2}\right)\right) ).
Steps to Solve
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Identify the Expression The expression given is ( 2\sin(t) + 3\cos(t) ).
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Write the Expression in a Standard Form We can try to express the sum ( 2\sin(t) + 3\cos(t) ) in the form ( R\sin(t + \phi) ). Here, ( R ) and ( \phi ) are constants that we need to find.
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Calculate ( R ) To find ( R ), we use the formula: $$ R = \sqrt{a^2 + b^2} $$ where ( a = 2 ) and ( b = 3 ): $$ R = \sqrt{2^2 + 3^2} = \sqrt{4 + 9} = \sqrt{13}. $$
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Determine ( \phi ) Now we calculate the angle ( \phi ) using: $$ \tan(\phi) = \frac{b}{a} = \frac{3}{2}. $$
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Express in Sine Form Thus, we rewrite the expression: $$ 2\sin(t) + 3\cos(t) = \sqrt{13}\sin(t + \phi) $$ where ( \phi = \tan^{-1}\left(\frac{3}{2}\right) ).
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Final Expression The final expression is: $$ 2\sin(t) + 3\cos(t) = \sqrt{13} \sin\left(t + \tan^{-1}\left(\frac{3}{2}\right)\right). $$
The simplified expression is ( \sqrt{13} \sin\left(t + \tan^{-1}\left(\frac{3}{2}\right)\right) ).
More Information
This transformation is useful in various applications, such as solving trigonometric equations or analyzing oscillatory motions. The angle ( \phi ) essentially represents a phase shift in the sine function.
Tips
- A common mistake is failing to calculate ( R ) correctly, particularly mixing up the values of ( a ) and ( b ).
- Not simplifying the angle ( \phi ) can lead to inaccuracies when rewriting the expression.
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