Find [tex]$\sin (A)$[/tex] if [tex]$\cos (A)=0.4$[/tex]
Round to the nearest ten-thousandth (four decimal places).

Use the Pythagorean identity sin 2 ( A ) + cos 2 ( A ) = 1 .
Substitute cos ( A ) = 0.4 into the identity and solve for sin 2 ( A ) .
Take the square root of sin 2 ( A ) to find sin ( A ) , considering both positive and negative roots.
Round the positive root to four decimal places: 0.9165 ​ .

Explanation

Problem Analysis We are given that cos ( A ) = 0.4 , and we need to find sin ( A ) . We will use the Pythagorean identity to relate sin ( A ) and cos ( A ) .

State Pythagorean Identity The Pythagorean identity states that sin 2 ( A ) + cos 2 ( A ) = 1

Substitute the value of cos(A) Substitute the given value of cos ( A ) into the identity: sin 2 ( A ) + ( 0.4 ) 2 = 1 sin 2 ( A ) + 0.16 = 1

Solve for sin^2(A) Solve for sin 2 ( A ) :
sin 2 ( A ) = 1 − 0.16 sin 2 ( A ) = 0.84

Take the square root Take the square root of both sides to find sin ( A ) . Remember that taking the square root can result in both positive and negative values: sin ( A ) = ± 0.84 ​

Calculate sin(A) Since the problem does not specify the quadrant of angle A , we consider both positive and negative roots. However, without further information, we will assume that sin ( A ) is positive. Calculate the positive square root: sin ( A ) = 0.84 ​ ≈ 0.916515138991168 Rounding to the nearest ten-thousandth (four decimal places), we get: sin ( A ) ≈ 0.9165

Final Answer Therefore, the value of sin ( A ) rounded to four decimal places is 0.9165.


Examples
Understanding trigonometric functions like sine and cosine is crucial in many real-world applications. For instance, when analyzing the motion of a pendulum, the sine function helps describe the horizontal displacement of the pendulum bob over time. Similarly, in electrical engineering, the cosine function is used to model alternating current (AC) waveforms. Knowing how to find the sine of an angle when you know its cosine allows engineers to predict and control the behavior of these systems, ensuring efficient and reliable performance.

Answered by GinnyAnswer | 2025-07-05

Using the Pythagorean identity, we determined that sin ( A ) equals approximately 0.9165 when cos ( A ) is 0.4. We arrived at this by substituting the value of cos ( A ) into the identity and solving for sin ( A ) . The final result is rounded to four decimal places: 0.9165.
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Answered by Anonymous | 2025-07-17