The simplified form, with no quotients, of $(10+\sin t)^2+\cos ^2 t$ is $\square$ .
Answer (1)
Expand the square: ( 10 + sin t ) 2 = 100 + 20 sin t + sin 2 t .
Substitute the expanded form into the original expression: 100 + 20 sin t + sin 2 t + cos 2 t .
Use the trigonometric identity sin 2 t + cos 2 t = 1 .
Simplify the expression to 101 + 20 sin t , so the final answer is 101 + 20 sin t .
Explanation
Understanding the Problem We are asked to simplify the expression ( 10 + sin t ) 2 + cos 2 t . Our goal is to expand and simplify the expression using trigonometric identities.
Expanding the Square First, we expand the square: ( 10 + sin t ) 2 = 1 0 2 + 2 ⋅ 10 ⋅ sin t + sin 2 t = 100 + 20 sin t + sin 2 t .
Substituting Back Now, we substitute the expanded form back into the original expression: ( 10 + sin t ) 2 + cos 2 t = ( 100 + 20 sin t + sin 2 t ) + cos 2 t = 100 + 20 sin t + sin 2 t + cos 2 t .
Using Trigonometric Identity We use the trigonometric identity sin 2 t + cos 2 t = 1 to simplify the expression: 100 + 20 sin t + sin 2 t + cos 2 t = 100 + 20 sin t + 1 = 101 + 20 sin t .
Final Answer Therefore, the simplified form of the expression is 101 + 20 sin t .
Examples
Understanding trigonometric identities and algebraic simplification is crucial in various fields such as physics, engineering, and computer graphics. For example, when analyzing wave phenomena in physics, simplifying expressions involving trigonometric functions helps in understanding the behavior of waves. In computer graphics, simplifying such expressions can optimize rendering processes, making them more efficient.
