Questions: Use table for trigonometric function values of some common angles and simplify the resulting expression. sin 45° cos 30° + cos 60° sin 60°

Transcript text: Use table for trigonometric function values of some common angles and simplify the resulting expression. \[ \sin 45^{\circ} \cos 30^{\circ}+\cos 60^{\circ} \sin 60^{\circ} \]

Solution

Solution Steps

Step 1: Identify Known Trigonometric Values

We start by identifying the known trigonometric values for the angles involved:

\(\sin 45^\circ = \frac{\sqrt{2}}{2} \approx 0.7071\)
\(\cos 30^\circ = \frac{\sqrt{3}}{2} \approx 0.8660\)
\(\cos 60^\circ = \frac{1}{2} = 0.5\)
\(\sin 60^\circ = \frac{\sqrt{3}}{2} \approx 0.8660\)

Step 2: Substitute Values into the Expression

Substitute these values into the given expression: \[ \sin 45^\circ \cos 30^\circ + \cos 60^\circ \sin 60^\circ \] \[ = \left(\frac{\sqrt{2}}{2}\right) \left(\frac{\sqrt{3}}{2}\right) + \left(\frac{1}{2}\right) \left(\frac{\sqrt{3}}{2}\right) \]

Step 3: Simplify the Expression

Simplify the expression by performing the multiplications: \[ = \frac{\sqrt{6}}{4} + \frac{\sqrt{3}}{4} \]

Combine the terms: \[ = \frac{\sqrt{6} + \sqrt{3}}{4} \]

Step 4: Approximate the Result

Calculate the approximate value of the expression: \[ \approx \frac{2.4495 + 1.7321}{4} \approx \frac{4.1816}{4} \approx 1.0454 \]