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. Step 1 A unit circle is shown. A ray intersects point (negative 2, negative 10) in quadrant 3. Theta is the angle formed by the ray and the x-axis in quadrant 1. Step 2 r = StartRoot (negative 2) squared + (negative 10) squared EndRoot = StartRoot 104 EndRoot = 2 StartRoot 26 EndRoot Step 3 sine theta = StartFraction negative 2 Over 2 StartRoot 26 EndRoot EndFraction = Negative StartFraction 1 Over StartRoot 26 EndRoot EndFraction = Negative StartFraction StartRoot 26 EndRoot Over 26 EndFraction cosine theta = StartFraction negative 10 Over 2 StartRoot 26 EndRoot EndFraction = Negative StartFraction 5 Over StartRoot 26 EndRoot EndFraction = Negative StartFraction 5 StartRoot 26 EndRoot Over 26 EndFraction tangent theta = StartFraction negative 2 Over negative 10 EndFraction = one-fifth cosecant theta = StartFraction 1 Over sine theta EndFraction = StartStartFraction 1 OverOver StartFraction Negative StartRoot 26 EndRoot Over 26 EndFRaction EndEndFraction = StartFraction negative StartRoot 26 EndRoot Over 5 EndFraction Secant theta = StartFraction 1 Over cosine theta EndFraction = StartStartFraction 1 OverOver Negative StartFraction 5 StartRoot 26 EndRoot Over 26 EndFraction EndEndFraction = StartFraction negative StartRoot 26 EndRoot Over 5 EndFraction cotangent theta = StartFraction 1 Over tangent theta EndFraction = StartFraction 1 Over one-fifth EndFraction = 5 Which of the following explains whether all of Francesca’s work is correct? Each step is correct because she plotted the point, drew a line to the x-axis to form a right triangle, used the Pythagorean theorem to find the hypotenuse, and finally wrote the correct ratios for all six functions. She made her first error in step 1 because she should have drawn the line to the y-axis to form the right triangle. She made her first error in step 2 because she should have used a negative value for r. She made her first error in step 3 because the sine, cosine, and tangent ratios are incorrect, which also results in incorrect cosecant, secant, and tangent functions.

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Answer:

STEP 3

Step-by-step explanation:

Francesca drew point (–2, –10) on the terminal ray of angle [tex]\Theta[/tex], which is in standard position. She found values for the six trigonometric functions using the steps below.

Step 1

A unit circle is shown. A ray intersects point (negative 2, negative 10) in quadrant 3. Theta is the angle formed by the ray and the x-axis in quadrant 1.

Step 2

[tex]r = (\sqrt{(-2)^2+(-10)^2}=\sqrt{104}=2\sqrt{26[/tex]

Step 3

[tex]Sin \theta = \frac{-2}{2\sqrt{26} }=-\frac{1}{\sqrt{26} }=-\frac{\sqrt{26}}{26 }[/tex]

[tex]cos \theta = \frac{-10}{2\sqrt{26} }=-\frac{5}{\sqrt{26} }=-\frac{5\sqrt{26}}{26 }[/tex]

[tex]tan \theta = \frac{2}{-10 }=-\frac{1}{5}\\[/tex]

[tex]cosec \theta = \frac{1}{sin \theta}=\frac{1}{-\frac{\sqrt{26}}{26 }}=-\frac{\sqrt{26}}{5}[/tex]

[tex]Secant \theta = 1/cos \theta =\frac{1}{-\frac{5\sqrt{26} }{26} } =-\frac{\sqrt{26}}{5} \\cotangent \theta=1/tan \theta=\frac{1}{1/5} =5[/tex]

Francesca made her first error in step 3 because the sine, cosine, and tangent ratios are incorrect, which also resulted in incorrect cosecant, secant, and tangent functions.

The correct values are:

[tex]Sin \theta = \frac{-10}{2\sqrt{26} }=-\frac{5}{\sqrt{26} }=-\frac{5\sqrt{26} }{26}\\cos \theta = \frac{-2}{2\sqrt{26} }=-\frac{1}{\sqrt{26} }=-\frac{\sqrt{26} }{26}\\tan \theta = \frac{-10}{-2}=5[/tex]

Answer:

D.She made her first error in step 3 because the sine, cosine, and tangent ratios are incorrect, which also results in incorrect cosecant, secant, and tangent functions.

Step-by-step explanation:

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