Give an exact answer for the other function values for [tex]\theta[/tex].
[tex]\cos \theta=-\frac{4 \sqrt{3}}{7}[/tex]
[tex]\tan \theta=-\frac{\sqrt{3}}{12}[/tex]
[tex]\cot \theta=-4 \sqrt{3}[/tex]
[tex]\sec \theta=\square[/tex]
[tex]\csc \theta=\square[/tex]
Answer (1)
Find sec θ using the reciprocal of cos θ : sec θ = c o s θ 1 = − 4 3 7 = − 12 7 3 .
Find csc 2 θ using the identity csc 2 θ = 1 + cot 2 θ : csc 2 θ = 1 + ( − 4 3 ) 2 = 49 .
Determine the sign of csc θ based on the quadrant of θ . Since cos θ < 0 and tan θ < 0 , θ is in the second quadrant, so 0"> csc θ > 0 .
Therefore, csc θ = 7 , and the final answers are − 12 7 3 and 7 .
Explanation
Understanding the Problem We are given cos θ = − 7 4 3 and tan θ = − 12 3 . We also know that cot θ = − 4 3 . Our goal is to find the exact values of sec θ and csc θ .
Finding sec(θ) First, we find sec θ using the identity sec θ = c o s θ 1 . Since cos θ = − 7 4 3 , we have sec θ = − 7 4 3 1 = − 4 3 7 To rationalize the denominator, we multiply the numerator and denominator by 3 :
sec θ = − 4 3 3 7 3 = − 4 ( 3 ) 7 3 = − 12 7 3 Thus, sec θ = − 12 7 3 .
Finding csc(θ) Next, we find csc θ . We know that cot θ = − 4 3 . We can use the identity csc 2 θ = 1 + cot 2 θ to find csc θ .
csc 2 θ = 1 + ( − 4 3 ) 2 = 1 + 16 ( 3 ) = 1 + 48 = 49 Taking the square root of both sides, we get csc θ = ± 7 .
Determining the Sign of csc(θ) Since tan θ = − 12 3 < 0 and cos θ = − 7 4 3 < 0 , θ is in the third quadrant is not possible, since in the third quadrant both tan and cos are positive. Thus, θ is in the second quadrant, where 0"> sin θ > 0 and 0"> csc θ > 0 . Therefore, csc θ = 7 .
Final Answer Therefore, we have sec θ = − 12 7 3 and csc θ = 7 .
Examples
Understanding trigonometric functions like cosine, tangent, secant, and cosecant is crucial in fields like navigation and surveying. For instance, surveyors use angles and distances to determine locations and boundaries. If a surveyor knows the cosine of an angle and needs to find the secant, they can use the reciprocal relationship between cosine and secant to calculate it. Similarly, cosecant is used in calculating heights and depths when the angle and opposite side are known.
