Question

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.

Answer 1

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.

Explanation:

on e 2020

Answer 2

STEP 3

Explanation:

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 = (√((-2)^2+(-10)^2)=√(104)=2\sqrt{26`

Step 3

`Sin \theta = (-2)/(2√(26) )=-(1)/(√(26) )=-(√(26))/(26 )`

`cos \theta = (-10)/(2√(26) )=-(5)/(√(26) )=-(5√(26))/(26 )`

`tan \theta = (2)/(-10 )=-(1)/(5)\\`

`cosec \theta = (1)/(sin \theta)=(1)/(-(√(26))/(26 ))=-(√(26))/(5)`

`Secant \theta = 1/cos \theta =(1)/(-(5√(26) )/(26) ) =-(√(26))/(5) \\cotangent \theta=1/tan \theta=(1)/(1/5) =5`

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:

`Sin \theta = (-10)/(2√(26) )=-(5)/(√(26) )=-(5√(26) )/(26)\\cos \theta = (-2)/(2√(26) )=-(1)/(√(26) )=-(√(26) )/(26)\\tan \theta = (-10)/(-2)=5`