a) Give an exact answer for the other function values for [tex]$\theta$[/tex].
[tex]$\cos \theta=-\frac{4 \sqrt{3}}{7}$[/tex]
(Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.)
[tex]$\tan \theta=-\frac{\sqrt{3}}{12}$[/tex]
[tex]$\cot \theta=-4 \sqrt{3}$[/tex]
(Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.)
[tex]$\sec \theta=-\frac{7 \sqrt{3}}{12}$[/tex]
[tex]$\csc \theta=7$[/tex]
(Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.)
b) Give an exact answer for each of the six function values for [tex]$\frac{\pi}{2}-\theta$[/tex].
[tex]$\sin \left(\frac{\pi}{2}-\theta\right)=\square \cos \left(\frac{\pi}{2}-\theta\right)=\square$[/tex]
(Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.)
Answer (2)
Determine sin θ using the Pythagorean identity sin 2 θ + cos 2 θ = 1 , resulting in sin θ = ± 7 1 .
Determine the sign of sin θ using the given tan θ and cos θ , concluding that sin θ = 7 1 .
Apply cofunction identities to find the trigonometric function values for 2 π − θ .
State the final answers: sin θ = 7 1 , sin ( 2 π − θ ) = − 7 4 3 , and cos ( 2 π − θ ) = 7 1 . sin θ = 7 1 , sin ( 2 π − θ ) = − 7 4 3 , cos ( 2 π − θ ) = 7 1
Explanation
Understanding the Problem We are given cos θ = − 7 4 3 , tan θ = − 12 3 , cot θ = − 4 3 , sec θ = − 12 7 3 , and csc θ = 7 . We need to find the value of sin θ and the six trigonometric function values for 2 π − θ .
Finding sin(theta) First, we need to find sin θ . We know that sin 2 θ + cos 2 θ = 1 . Substituting the given value of cos θ , we have sin 2 θ + ( − 7 4 3 ) 2 = 1
Solving for sin(theta) sin 2 θ + 49 16 × 3 = 1 sin 2 θ + 49 48 = 1 sin 2 θ = 1 − 49 48 sin 2 θ = 49 49 − 48 sin 2 θ = 49 1 Taking the square root of both sides, we get sin θ = ± 7 1
Determining the sign of sin(theta) Since tan θ = c o s θ s i n θ = − 12 3 and cos θ = − 7 4 3 is negative, sin θ must be positive because a negative divided by a negative is a positive. Therefore, sin θ = 7 1 .
Using Cofunction Identities Now, we need to find the six trigonometric function values for 2 π − θ . We use the cofunction identities: sin ( 2 π − θ ) = cos θ = − 7 4 3 cos ( 2 π − θ ) = sin θ = 7 1 tan ( 2 π − θ ) = cot θ = − 4 3 cot ( 2 π − θ ) = tan θ = − 12 3 sec ( 2 π − θ ) = csc θ = 7 csc ( 2 π − θ ) = sec θ = − 12 7 3
Final Answer Therefore, sin θ = 7 1 , sin ( 2 π − θ ) = − 7 4 3 and cos ( 2 π − θ ) = 7 1 .
Examples
Understanding trigonometric functions and their cofunctions is crucial in fields like physics and engineering. For example, when analyzing projectile motion, the initial velocity can be broken down into horizontal and vertical components using sine and cosine. The relationships between these components at complementary angles (like θ and 2 π − θ ) help engineers design trajectories for projectiles, ensuring they reach their intended target efficiently. This knowledge is also vital in signal processing, where understanding phase shifts and frequency components relies heavily on trigonometric identities.
We found that sin θ = 7 1 , and for 2 π − θ , sin ( 2 π − θ ) = − 7 4 3 and cos ( 2 π − θ ) = 7 1 . All other trigonometric values were also derived directly from the original trigonometric functions and identities.
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