QuestionFind the six trigonometric functions for the angle $765^{\circ}$ . Calculate $\sin 765^{\circ}=$ (simplify and rationalize).

Studdy Solution

STEP 1

Assumptions1. We are asked to find the exact values of the six trigonometric functions of the angle $765^{\circ}$ . . The six trigonometric functions are sine (sin), cosine (cos), tangent (tan), cosecant (csc), secant (sec), and cotangent (cot).
3. We know that the trigonometric functions repeat every $360^{\circ}$ , so we can reduce $765^{\circ}$ to an equivalent angle between $0^{\circ}$ and $360^{\circ}$ .

STEP 2

First, we need to find an equivalent angle to $765^{\circ}$ that is between $0^{\circ}$ and $360^{\circ}$ . We can do this by subtracting multiples of $360^{\circ}$ from $765^{\circ}$ .
$Equivalent\, angle =765^{\circ} -360^{\circ} \times n$ where $n$ is an integer such that the equivalent angle is between $0^{\circ}$ and $360^{\circ}$ .

STEP 3

Now, plug in the given value for the angle and find the appropriate value for $n$ .
$Equivalent\, angle =765^{\circ} -360^{\circ} \times2 =45^{\circ}$

STEP 4

Now that we have the equivalent angle, we can find the exact values of the six trigonometric functions for $45^{\circ}$ .

STEP 5

For $45^{\circ}$ , the values of the trigonometric functions are known $\sin45^{\circ} = \frac{\sqrt{2}}{2}$ $\cos45^{\circ} = \frac{\sqrt{2}}{2}$ $\tan45^{\circ} =1$ $\csc45^{\circ} = \sqrt{2}$ $\sec45^{\circ} = \sqrt{2}$ $\cot45^{\circ} =1$ So, the exact values of the six trigonometric functions for $765^{\circ}$ are the same as those for $45^{\circ}$ .

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