Keisha and David each found the same value for [tex]\cos \theta[/tex], as shown below, given [tex]\sin \theta=-\frac{8}{17}[/tex]
| Kersha's Solution | David's Solution |
|---|---|
| [tex]\begin{array}{c}
\frac{\left(-\frac{8}{17}\right)^2}{\cos ^2 \theta}+1=\frac{1}{\cos ^2 \theta} \
\left(-\frac{8}{17}\right)^2+\cos ^2 \theta=1 \
\cos \theta= \pm \sqrt{1-\frac{64}{289}}\end{array}[/tex] | [tex]\begin{array}{l}
\sin ^2 \theta+\cos ^2 \theta=1 \
\cos ^2 \theta=1-\left(-\frac{8}{17}\right)^2 \
\cos \theta= \pm \sqrt{\frac{225}{281}} \
\cos \theta= \pm \frac{15}{17}\end{array}[/tex] |
Answer (2)
Both Keisha and David correctly used trigonometric identities to find cos θ .
Keisha used the identity tan 2 θ + 1 = sec 2 θ , while David used sin 2 θ + cos 2 θ = 1 .
Both solutions lead to cos 2 θ = 1 − ( − 17 8 ) 2 = 289 225 .
Therefore, cos θ = ± 17 15 , and there is no error in either solution. cos θ = ± 17 15
Explanation
Problem Setup We are given that sin θ = − 17 8 , and we need to analyze Keisha's and David's solutions for finding cos θ to identify any errors.
Analyzing Keisha's Solution Keisha's solution starts with the equation c o s 2 θ ( − 17 8 ) 2 + 1 = c o s 2 θ 1 . Multiplying both sides by cos 2 θ , we get ( − 17 8 ) 2 + cos 2 θ = 1 . This leads to cos 2 θ = 1 − ( − 17 8 ) 2 = 1 − 289 64 = 289 289 − 64 = 289 225 . Therefore, cos θ = ± 289 225 = ± 17 15 .
Analyzing David's Solution David's solution uses the Pythagorean identity sin 2 θ + cos 2 θ = 1 . Substituting sin θ = − 17 8 , we have ( − 17 8 ) 2 + cos 2 θ = 1 , which gives cos 2 θ = 1 − ( − 17 8 ) 2 = 1 − 289 64 = 289 225 . Thus, cos θ = ± 289 225 = ± 17 15 .
Verifying Keisha's Initial Equation Both Keisha and David arrive at the same result: cos θ = ± 17 15 . However, the initial equation in Keisha's solution, c o s 2 θ ( − 17 8 ) 2 + 1 = c o s 2 θ 1 , seems unusual. This equation is derived from the identity tan 2 θ + 1 = sec 2 θ , where tan θ = c o s θ s i n θ and sec θ = c o s θ 1 . Substituting these into the identity, we get c o s 2 θ s i n 2 θ + 1 = c o s 2 θ 1 . This is exactly the equation Keisha used. So, Keisha's initial equation is correct.
Determining the Correct Value of Cosine Since sin θ = − 17 8 is negative, θ must be in either the third or fourth quadrant. In the third quadrant, both sin θ and cos θ are negative. In the fourth quadrant, sin θ is negative and cos θ is positive. Therefore, cos θ can be either 17 15 or − 17 15 . Both Keisha and David correctly found both possible values for cos θ . There is no error in either solution.
Examples
Understanding trigonometric identities and how to apply them is crucial in fields like physics and engineering. For example, when analyzing the motion of a pendulum or the behavior of alternating current in an electrical circuit, trigonometric functions are used to model oscillations and wave patterns. Knowing the relationships between sine, cosine, and tangent allows engineers to predict and control these systems effectively.
Both Keisha and David correctly computed cos θ using trigonometric identities and arrived at cos θ = ± 17 15 . Both methods demonstrated the proper use of the Pythagorean identity. Their answers differ in signs based on the quadrant in which θ lies.
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