Correct Answer
sine of theta equals 0 point 6 or equivalent
$$ \sin(\theta)=0.6, \text{or equivalent} $$
sine of pi plus theta equals negative 0 point 6 or equivalent $$ \sin(\pi+\theta)=−0.6, \text{or equivalent} $$
Typical equivalent responses include three fifths and negative three fifths, \( \frac{3}{5} \text{ and } -\frac{3}{5} \) or 6 tenths and negative 6 tenths. \( \frac{6}{10} \text{ and } -\frac{6}{10}. \)
Explanation
You might recognize this as a scaling up of the common 3-4-5 right triangle. Sides are multiples of 3, 4, and 5: 6, 8, and 10. If you didn't recognize that the hypotenuse was 10, you could find it using the Pythagorean Theorem:a squared plus b squared equals c squared
$$ a^2 + b^2 = c^2 $$
c equals the square root of a squared plus b squared, which equals the square root of 6 squared plus 8 squared, which equals the square root of 36 plus 64, which equals the square root of 100, which equals 10. $$ c = \sqrt{a^2 + b^2} = \sqrt{6^2 + 8^2} = \sqrt{36+64} = \sqrt{100} = 10. $$
The sine of an angle is the length of the side opposite that angle divided by the length of the hypotenuse. The side opposite theta \( \theta \) has a length of 6, and the hypotenuse has a length of 10, which gives us sine of theta equals six tenths. \( \sin(\theta)=\frac{6}{10}. \)
Given that sine of theta equals 3 fifths, \( \sin(\theta) = \frac{3}{5}, \) we can use our understanding of the symmetry and periodicity of the sine function on the unit circle to deduce sine of pi plus theta. \( \sin(\pi + \theta). \)
The angle theta \( \theta \) shown on the graph is in the first quadrant. It has a positive y value and thus a positive sine. By adding pi \( \pi \) (180°) to that angle, we move it into the third quadrant. On the unit circle, this angle has a negative y but with the same absolute value as angle theta. \( \theta. \) In the third quadrant open parenthesis pi plus theta, \( ( \pi + \theta ), \) sine is negative, and \( \sin(\pi + \theta) = -\sin(\theta) \). By applying these properties, we can determine that sine of pi plus theta equals negative sine of theta \( \sin(\pi + \theta) = -\sin(\theta), \) which led to the final result:sine of pi plus theta equals negative 3 fifths.
$$ \sin(\pi + \theta) = -\frac{3}{5} $$
Scoring Note
Although this question actually asks two questions, it is worth only one point, and there is no partial credit for getting only one part correct. In fact, there are no partial credit questions on the MCAP math tests, except for the constructed-response questions that are scored using a holistic scoring rubric.
Furthermore, although we would advise making a direct response, questions like this are scored using an artificial intelligence engine. Any value in the box, either fractional or decimal, that evaluates to the correct answer would be counted as correct. Leading or trailing zeroes, as well as commas, are typically ignored by the scoring engine. As long as what is typed in evaluates to the correct number, credit will be given.
