Math

QuestionFind the six trigonometric functions of tt for the point (35,225)\left(-\frac{\sqrt{3}}{5},-\frac{\sqrt{22}}{5}\right) on the unit circle.

Studdy Solution

STEP 1

Assumptions1. The point (35,225)\left(-\frac{\sqrt{3}}{5},-\frac{\sqrt{22}}{5}\right) lies on the unit circle. . The point corresponds to a real number tt.
3. We need to find the six trigonometric functions of tt: sin(t)\sin(t), cos(t)\cos(t), tan(t)\tan(t), csc(t)\csc(t), sec(t)\sec(t), and cot(t)\cot(t).

STEP 2

On the unit circle, the x-coordinate corresponds to the cosine of the angle and the y-coordinate corresponds to the sine of the angle. So we can directly say thatcos(t)=5\cos(t) = -\frac{\sqrt{}}{5}sin(t)=225\sin(t) = -\frac{\sqrt{22}}{5}

STEP 3

Next, we can find the tangent of the angle using the formula tan(t)=sin(t)cos(t)\tan(t) = \frac{\sin(t)}{\cos(t)}.tan(t)=22535\tan(t) = \frac{-\frac{\sqrt{22}}{5}}{-\frac{\sqrt{3}}{5}}

STEP 4

implify the fraction to calculate the value of tan(t)\tan(t).
tan(t)=223=223\tan(t) = \frac{\sqrt{22}}{\sqrt{3}} = \sqrt{\frac{22}{3}}

STEP 5

The cosecant of the angle is the reciprocal of the sine, so csc(t)=1sin(t)\csc(t) = \frac{1}{\sin(t)}.
csc(t)=1225\csc(t) = \frac{1}{-\frac{\sqrt{22}}{5}}

STEP 6

implify the fraction to calculate the value of csc(t)\csc(t).
csc(t)=522=52222\csc(t) = -\frac{5}{\sqrt{22}} = -\frac{5\sqrt{22}}{22}

STEP 7

The secant of the angle is the reciprocal of the cosine, so sec(t)=1cos(t)\sec(t) = \frac{1}{\cos(t)}.
sec(t)=135\sec(t) = \frac{1}{-\frac{\sqrt{3}}{5}}

STEP 8

implify the fraction to calculate the value of sec(t)\sec(t).
sec(t)=53=533\sec(t) = -\frac{5}{\sqrt{3}} = -\frac{5\sqrt{3}}{3}

STEP 9

The cotangent of the angle is the reciprocal of the tangent, so cot(t)=tan(t)\cot(t) = \frac{}{\tan(t)}.
cot(t)=223\cot(t) = \frac{}{\sqrt{\frac{22}{3}}}

STEP 10

implify the fraction to calculate the value of cot(t)\cot(t).
cot(t)=322\cot(t) = \sqrt{\frac{3}{22}}So, the six trigonometric functions of tt aresin(t)=225\sin(t) = -\frac{\sqrt{22}}{5}cos(t)=35\cos(t) = -\frac{\sqrt{3}}{5}tan(t)=223\tan(t) = \sqrt{\frac{22}{3}}csc(t)=52222\csc(t) = -\frac{5\sqrt{22}}{22}sec(t)=533\sec(t) = -\frac{5\sqrt{3}}{3}cot(t)=322\cot(t) = \sqrt{\frac{3}{22}}

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