Find the exact values of the following:
(a) [tex]$\cos (\arctan (-10))$[/tex]
(b) [tex]$\cos \left(\arcsin \left(\frac{-8}{17}\right)\right.[/tex]
Answer (2)
For (a), let θ = arctan ( − 10 ) , then tan ( θ ) = − 10 , and using the Pythagorean identity, cos ( θ ) = 101 1 = 101 101 .
For (b), let ϕ = arcsin ( 17 − 8 ) , then sin ( ϕ ) = 17 − 8 , and using the Pythagorean identity, cos ( ϕ ) = 17 15 .
The exact value of cos ( arctan ( − 10 )) is 101 101 .
The exact value of cos ( arcsin ( 17 − 8 )) is 17 15 .
101 101 , 17 15
Explanation
Problem Overview We are asked to find the exact values of two expressions involving trigonometric functions and their inverses. Let's tackle them one by one.
Solving Part (a) (a) We need to find cos ( arctan ( − 10 )) . Let θ = arctan ( − 10 ) . This means that tan ( θ ) = − 10 . We want to find cos ( θ ) . Since tan ( θ ) = c o s ( θ ) s i n ( θ ) , we have sin ( θ ) = − 10 cos ( θ ) .
Using the Pythagorean identity sin 2 ( θ ) + cos 2 ( θ ) = 1 , we can substitute sin ( θ ) to get ( − 10 cos ( θ ) ) 2 + cos 2 ( θ ) = 1 . This simplifies to 100 cos 2 ( θ ) + cos 2 ( θ ) = 1 , so 101 cos 2 ( θ ) = 1 .
Thus, cos 2 ( θ ) = 101 1 , so cos ( θ ) = ± 101 1 . Since the range of arctan ( x ) is ( − 2 π , 2 π ) , and tan ( θ ) = − 10 < 0 , we know that − 2 π < θ < 0 . In this interval, 0"> cos ( θ ) > 0 . Therefore, cos ( θ ) = 101 1 = 101 101 .
Solving Part (b) (b) We need to find cos ( arcsin ( 17 − 8 )) . Let ϕ = arcsin ( 17 − 8 ) . This means that sin ( ϕ ) = 17 − 8 . We want to find cos ( ϕ ) .
Using the Pythagorean identity sin 2 ( ϕ ) + cos 2 ( ϕ ) = 1 , we have ( 17 − 8 ) 2 + cos 2 ( ϕ ) = 1 , which simplifies to 289 64 + cos 2 ( ϕ ) = 1 .
Thus, cos 2 ( ϕ ) = 1 − 289 64 = 289 289 − 64 = 289 225 , so cos ( ϕ ) = ± 289 225 = ± 17 15 . Since the range of arcsin ( x ) is [ − 2 π , 2 π ] , and sin ( ϕ ) = 17 − 8 < 0 , we know that − 2 π ≤ ϕ < 0 . In this interval, 0"> cos ( ϕ ) > 0 . Therefore, cos ( ϕ ) = 17 15 .
Final Answer Therefore, the exact values are: (a) cos ( arctan ( − 10 )) = 101 101 (b) cos ( arcsin ( 17 − 8 )) = 17 15
Examples
Understanding trigonometric functions and their inverses is crucial in various fields, such as physics and engineering. For example, when analyzing the trajectory of a projectile, you might need to find the angle at which it was launched given its initial velocity components. This involves using inverse trigonometric functions like arctan to determine the angle from the ratio of the velocity components. Similarly, in electrical engineering, calculating the phase angle between voltage and current in an AC circuit often requires using arctan. These calculations help engineers design and optimize systems for maximum efficiency and safety.
The exact values are cos ( arctan ( − 10 )) = 101 101 and cos ( arcsin ( 17 − 8 )) = 17 15 .
;
