Question - Trigonometric Values of Special Angles
Solution:
\begin{align*}\sin(45^\circ) &= \frac{\sqrt{2}}{2},\\\cos(45^\circ) &= \frac{\sqrt{2}}{2},\\\tan(45^\circ) &= \frac{\sin(45^\circ)}{\cos(45^\circ)} = \frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} = 1.\end{align*}
Tutor Solution:
To find the exact values of the trigonometric functions at $$45^\circ$$, we can use the properties of an isosceles right-angled triangle, where the two non-hypotenuse sides are equal. This implies that the two angles other than the right angle are also equal and thus both are $$45^\circ$$.In such a triangle, if we consider the two equal sides to be of length 1, then by the Pythagorean theorem, the hypotenuse (diagonal) would be of length $$\sqrt{2}$$.The sine of an angle in a right-angled triangle is defined as the ratio of the opposite side to the hypotenuse. So for $$45^\circ$$:$$\sin(45^\circ) = \frac{\text{opposite side}}{\text{hypotenuse}} = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}$$Similarly, the cosine of an angle is the ratio of the adjacent side to the hypotenuse. For $$45^\circ$$:$$\cos(45^\circ) = \frac{\text{adjacent side}}{\text{hypotenuse}} = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}$$Note that here, both the sine and cosine of $$45^\circ$$ are equal due to the isosceles nature of the triangle.Finally, the tangent of an angle is the ratio of the opposite side to the adjacent side. For $$45^\circ$$:$$\tan(45^\circ) = \frac{\text{opposite side}}{\text{adjacent side}} = \frac{1}{1} = 1$$Therefore, the exact values for the trigonometric functions at $$45^\circ$$ are:$$\sin(45^\circ) = \frac{\sqrt{2}}{2}$$$$\cos(45^\circ) = \frac{\sqrt{2}}{2}$$$$\tan(45^\circ) = 1$$These are the exact values for the sine, cosine, and tangent of $$45^\circ$$.
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