Consider the relationship, given pi over 2 is less than theta comma theta is less than pi sine squared theta plus cosine squared theta is equal to 1 Which of the following best explains how this relationship and the value of sine theta can be used to find the other trigonometric values?
Answer
A
The values of sine theta and cosine theta represent the legs of a right triangle with a hypotenuse of negative 1 , since theta is in Quadrant II; therefore, solving for cosine theta finds the unknown leg, and then all other trigonometric values can be found.
B
The values of sine theta and cosine theta 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.
C
The values of sine theta and cosine theta 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.
D
The values of sine theta and cosine theta represent the legs of a right triangle with a hypotenuse of 1 ; therefore, solving for cosine theta finds the unknown leg, and then all other trigonometric values can be found.
Explanation
The identity sin²θ + cos²θ = 1 matches the Pythagorean theorem for a right triangle whose hypotenuse is 1. If sin θ is known, substitute it into the identity and solve for cos θ; because θ is in Quadrant II, cos θ must be negative. The sine and cosine values can then be used to find the other trigonometric ratios. Choice D correctly identifies the hypotenuse as 1; a hypotenuse cannot be negative as in choice A.
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