Consider the relationship below, given [tex]$\frac{\pi}{2}\ \textless \ \theta\ \textless \ \pi$[/tex].
[tex]$\sin ^2 \theta+\cos ^2 \theta=1$[/tex]
Which of the following best explains how this relationship and the value of [tex]$\sin \theta$[/tex] can be used to find the other trigonometric values?
A. The values of [tex]$\sin \theta$[/tex] and [tex]$\cos \theta$[/tex] represent the legs of a right triangle with a hypotenuse of 1; therefore, solving for [tex]$\cos \theta$[/tex] finds the unknown leg, and then all other trigonometric values can be found.
B. The values of [tex]$\sin \theta$[/tex] and [tex]$\cos \theta$[/tex] represent the angles of a right triangle; therefore, solving the relationship will find all three angles of the triangle, and then all trigonometric values can be found.
C. The values of [tex]$\sin \theta$[/tex] and [tex]$\cos \theta$[/tex] represent the angles of a right triangle; therefore, other pairs of trigonometric ratios will have the same sum, 1, which can then be used to find all other values.
D. The values of [tex]$\sin \theta$[/tex] and [tex]$\cos \theta$[/tex] represent the legs of a right triangle with a hypotenuse of -1, since [tex]$\theta$[/tex] is in Quadrant II; therefore, solving for [tex]$\cos \theta$[/tex] finds the unknown leg, and then all other trigonometric values can be found.
Answer (2)
Use the identity sin 2 θ + cos 2 θ = 1 to find cos θ given sin θ .
Consider the quadrant to determine the sign of cos θ .
Interpret sin θ and cos θ as legs of a right triangle with hypotenuse 1.
The correct explanation is: The values of sin θ and cos θ represent the legs of a right triangle with a hypotenuse of 1; therefore, solving for cos θ finds the unknown leg, and then all other trigonometric values can be found.
Explanation
Analyze the problem and given data The problem states that we are given the trigonometric identity sin 2 θ + cos 2 θ = 1 and that 2 π < θ < π . This means that θ is in the second quadrant. We are also told that we know the value of sin θ and we need to determine how this information can be used to find the other trigonometric values.
Explain how to find other trigonometric values The identity sin 2 θ + cos 2 θ = 1 relates sin θ and cos θ . If we know sin θ , we can solve for cos θ using this identity. Since θ is in the second quadrant, cos θ will be negative. Once we have both sin θ and cos θ , we can find the other trigonometric functions: tan θ = c o s θ s i n θ , csc θ = s i n θ 1 , sec θ = c o s θ 1 , and cot θ = t a n θ 1 .
Relate sine and cosine to a right triangle The values of sin θ and cos θ can be thought of as the vertical and horizontal coordinates, respectively, of a point on the unit circle. They also represent the legs of a right triangle with a hypotenuse of 1. Solving for cos θ finds the length of the adjacent side (x-coordinate), and then all other trigonometric values can be found using the definitions of the trigonometric functions.
State the final answer Based on the analysis, the best explanation is: The values of sin θ and cos θ represent the legs of a right triangle with a hypotenuse of 1; therefore, solving for cos θ finds the unknown leg, and then all other trigonometric values can be found.
Examples
Consider a scenario where you're navigating a sailboat. Knowing the angle at which the wind hits your sail ( θ ) and the sine of that angle, you can determine the cosine using the trigonometric identity sin 2 t h e t a + co s 2 t h e t a = 1 . This allows you to calculate the force component pushing the boat forward, demonstrating a practical application of trigonometric relationships in sailing and navigation.
The identity sin 2 θ + cos 2 θ = 1 indicates a relationship between sine and cosine in a right triangle with a hypotenuse of 1. Knowing sin θ allows the calculation of cos θ and other trigonometric values. This leads to the conclusion that the correct answer is option A.
;
