Check all that apply. If [tex]$\csc \theta=\frac{13}{5}$[/tex], then:
A. [tex]$\tan \theta=\frac{5}{12}$[/tex]
B. [tex]$\sin \theta=\frac{5}{13}$[/tex]
C. [tex]$\cos \theta=\frac{5}{13}$[/tex]
D. [tex]$\sec \theta=\frac{5}{13}$[/tex]
Answer (1)
We are given csc θ = 5 13 .
We find sin θ = c s c θ 1 = 13 5 .
We use sin 2 θ + cos 2 θ = 1 to find cos θ = ± 13 12 .
We find tan θ = c o s θ s i n θ = ± 12 5 .
A , B
Explanation
Analyze the given information We are given that csc θ = 5 13 . We need to determine which of the given options are correct.
Calculate sin θ Since csc θ = 5 13 , we can find sin θ using the identity sin θ = c s c θ 1 . Therefore, sin θ = 5 13 1 = 13 5 So, option B is correct.
Calculate cos θ Now, we use the Pythagorean identity sin 2 θ + cos 2 θ = 1 to find cos θ .
( 13 5 ) 2 + cos 2 θ = 1 169 25 + cos 2 θ = 1 cos 2 θ = 1 − 169 25 = 169 169 − 25 = 169 144 cos θ = ± 169 144 = ± 13 12
Calculate tan θ Next, we find tan θ using the identity tan θ = c o s θ s i n θ .
tan θ = ± 13 12 13 5 = ± 12 5 So, option A, tan θ = 12 5 , can be correct if cos θ = 13 12 .
Calculate sec θ Finally, we find sec θ using the identity sec θ = c o s θ 1 .
sec θ = ± 13 12 1 = ± 12 13 So, option D, sec θ = 13 5 , is incorrect. Also, option C, cos θ = 13 5 , is incorrect.
Conclusion Therefore, the correct options are A and B.
Examples
Understanding trigonometric functions like sine, cosine, tangent, cosecant, secant, and cotangent is crucial in various fields. For instance, in navigation, these functions help determine the direction and distance of a ship or aircraft. In physics, they are used to analyze wave phenomena, such as light and sound waves. In engineering, trigonometric functions are essential for designing structures, bridges, and electrical circuits. Knowing how these functions relate to each other and how to calculate their values for different angles allows us to solve real-world problems involving angles and distances.
