Type the correct answer in each box. Use numerals instead of words. If necessary, use / for the fraction bar(s).
A right triangle [tex]$A B C$[/tex] has complementary angles [tex]$A$[/tex] and [tex]$C$[/tex].
If [tex]$\sin (A)=\frac{24}{25}$[/tex], the value of [tex]$\cos (C)=$\square[/tex],
If [tex]$\cos (C)=\frac{20}{29}$[/tex], the value of [tex]$\sin (A)=$\square[/tex]
Answer (1)
Recognizes that in a right triangle, angles A and C are complementary, implying sin ( A ) = cos ( C ) .
Given sin ( A ) = 25 24 , concludes that cos ( C ) = 25 24 .
Given cos ( C ) = 29 20 , determines that sin ( A ) = 29 20 .
States the final answers based on the complementary relationship: 25 24 , 29 20 .
Explanation
Understanding Complementary Angles In a right triangle, the two acute angles are complementary. This means that the sine of one acute angle is equal to the cosine of the other acute angle. In other words, if angles A and C are complementary, then sin ( A ) = cos ( C ) and cos ( A ) = sin ( C ) .
Finding cos(C) Given that sin ( A ) = 25 24 , we can directly conclude that cos ( C ) = 25 24 because A and C are complementary angles in a right triangle.
Finding sin(A) Similarly, given that cos ( C ) = 29 20 , we can directly conclude that sin ( A ) = 29 20 because A and C are complementary angles in a right triangle.
Final Answer Therefore, the value of cos ( C ) is 25 24 and the value of sin ( A ) is 29 20 .
Examples
Understanding complementary angles is useful in various real-life scenarios, such as calculating the trajectory of projectiles or designing structures where angles play a critical role. For example, when building a ramp, knowing the sine and cosine of the angles helps determine the slope and stability of the ramp. If you know one angle, you automatically know the other angle's sine or cosine, making calculations easier: sin ( A ) = cos ( 90 − A ) .
