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Chapter 6: Problem 66

Use the half-angle formulas to simplify the expression. $$\sqrt{\frac{1+\cos 4 x}{2}}$$

Short Answer

Expert verified
The simplified expression of \(\sqrt{\frac{1+\cos 4 x}{2}}\) using the half-angle formula is \(\cos (2x)\).

Step by step solution

01

Recall Half-Angle Formula

Recall the half-angle formula for cosine which is \(\cos (\theta/2) = \pm \sqrt{\frac{1+\cos \theta}{2}}\). This will be used to simplify the given expression.
02

Compare Given Expression with Half-Angle Formula

Notice that the given expression is in the form of the right-hand side of the half-angle identity. With \(\theta = 4x\) we have: \(4x/2 = 2x\) which will be used as the simplified version of the given expression.
03

Apply Half-Angle Formula

Applying the half-angle formula directly results in: \(\cos (2x)\).

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Most popular questions from this chapter

Use the half-angle formulas to determine the exact values of the sine, cosine, and tangent of the angle. $$75^{\circ}$$

Find the exact values of \(\sin (u / 2), \cos (u / 2),\) and \(\tan (u / 2)\) using the half-angle formulas. $$\sin u=\frac{5}{13}, \quad \pi / 2

Verify the identity algebraically. Use a graphing utility to check your result graphically. $$\csc 2 \theta=\frac{\csc \theta}{2 \cos \theta}$$

Write the trigonometric expression as an algebraic expression. $$\cos (2 \arctan x)$$

Use the sum-to-product formulas to find the exact value of the expression. $$\sin 75^{\circ}+\sin 15^{\circ}$$
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