Check all that apply. If [tex]$\sin \theta=\frac{15}{17}$[/tex], then:
A. [tex]$\tan \theta=\frac{15}{8}$[/tex]
B. [tex]$\csc \theta=\frac{17}{15}$[/tex]
C. [tex]$\sec \theta=\frac{17}{8}$[/tex]
D. [tex]$\cos \theta=\frac{17}{8}$[/tex]
Answer (2)
We are given sin θ = 17 15 .
We find cos θ using the Pythagorean identity: cos θ = 1 − sin 2 θ = 17 8 .
We calculate tan θ = c o s θ s i n θ = 8 15 , csc θ = s i n θ 1 = 15 17 , and sec θ = c o s θ 1 = 8 17 .
The correct options are A, B, and C.
A , B , C
Explanation
Problem Analysis We are given that sin θ = 17 15 . We need to determine which of the given options are correct.
Calculate cos(theta) First, we find cos θ using the identity sin 2 θ + cos 2 θ = 1 . Thus, cos θ = 1 − sin 2 θ . Substituting the given value, we have cos θ = 1 − ( 17 15 ) 2 = 1 − 289 225 = 289 289 − 225 = 289 64 = 17 8 So, cos θ = 17 8 .
Calculate tan(theta) Next, we find tan θ using the identity tan θ = c o s θ s i n θ . Substituting the values, we have tan θ = 17 8 17 15 = 8 15 So, tan θ = 8 15 .
Calculate csc(theta) Now, we find csc θ using the identity csc θ = s i n θ 1 . Substituting the given value, we have csc θ = 17 15 1 = 15 17 So, csc θ = 15 17 .
Calculate sec(theta) Then, we find sec θ using the identity sec θ = c o s θ 1 . Substituting the value of cos θ , we have sec θ = 17 8 1 = 8 17 So, sec θ = 8 17 .
Compare and select correct options Comparing the calculated values with the given options: A. tan θ = 8 15 is correct. B. csc θ = 15 17 is correct. C. sec θ = 8 17 is correct. D. cos θ = 8 17 is incorrect, since we found cos θ = 17 8 .
Final Answer Therefore, the correct options are A, B, and C.
Examples
Understanding trigonometric relationships is crucial in fields like navigation and surveying. For instance, if a surveyor knows the angle of elevation to the top of a building and the distance to the building's base, they can use trigonometric functions like sine, cosine, and tangent to determine the building's height. Similarly, in navigation, sailors use these functions to calculate distances and directions based on angles measured from celestial bodies.
The correct answers are A, B, and C, as they confirm the calculated values for tangent, cosecant, and secant based on the given value of sine. The cosine value option D is incorrect, as cos θ = 17 8 .
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