Francesca drew point $(-2,-10)$ on the terminal ray of angle $\theta$, which is in standard position. She found values for the six trigonometric functions using the steps below.
Answer (2)
The problem involves finding the sine of an angle given a point on its terminal ray.
The radius r is calculated correctly as 2 26 .
The sine of the angle is calculated using the formula sin θ = r y .
The correct value for sin θ is found to be − 26 5 26 .
Explanation
Identify the problem The problem states that Francesca calculated the value of sin θ incorrectly. We need to identify and correct her mistake.
Calculate the radius The point ( − 2 , − 10 ) lies on the terminal ray of angle θ in standard position. The radius r is calculated using the formula r = x 2 + y 2 . In this case, x = − 2 and y = − 10 . So, r = ( − 2 ) 2 + ( − 10 ) 2 = 4 + 100 = 104 = 2 26 . This step is correct.
Identify the error The sine of θ is calculated using the formula sin θ = r y . Francesca incorrectly used x instead of y in her calculation. The correct calculation is sin θ = 2 26 − 10 = 26 − 5 .
Rationalize the denominator To rationalize the denominator, we multiply the numerator and denominator by 26 : sin θ = 26 − 5 ⋅ 26 26 = 26 − 5 26 .
State the correct value Therefore, the correct value for sin θ is − 26 5 26 .
Examples
Understanding trigonometric functions is crucial in fields like navigation and physics. For example, when a ship is sailing, the bearing (angle) and distance to a landmark can be used to calculate its position using trigonometric functions. Similarly, in physics, the trajectory of a projectile can be determined using trigonometric functions if the initial velocity and launch angle are known.
We found the sine of angle θ to be − 26 5 26 and derived the other trigonometric functions using the coordinates ( − 2 , − 10 ) . The radius was calculated as 2 26 . By using the correct formulas, all six trigonometric functions were determined accurately.
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