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Chapter 5: Problem 11

Find the exact value of each expression. (a) \(\sin \left(135^{\circ}-30^{\circ}\right)\) (b) \(\sin 135^{\circ}-\cos 30^{\circ}\)

Short Answer

Expert verified
The exact value of the expressions are (a) \( \sqrt{6}/4 + \sqrt{2}/4 \) and (b) \( \sqrt{2}/2 - \sqrt{3}/2 \)

Step by step solution

01

Simplify the First Expression

Solve the expression \(\sin \left(135^{\circ}-30^{\circ}\right)\) by using the trigonometric identity \(\sin(\alpha - \beta) = \sin\alpha\cos\beta - \cos\alpha\sin\beta\). Thus, the first expression becomes \(\sin 135^{\circ}\cos 30^{\circ} - \cos 135^{\circ}\sin 30^{\circ}\).
02

Substitute Known Trigonometric Values

Substitute the known values of sine and cosine for the respective angles: \(\sqrt{2}/2 \times \sqrt{3}/2 - -\sqrt{2}/2 \times 1/2 = \sqrt{6}/4 + \sqrt{2}/4\). Similar computation is done for the second term as well, and the result is achieved.
03

Simplify the Second Expression

The expression \(\sin 135^{\circ}-\cos 30^{\circ}\) is straightforward. It requires substituting the known trigonometric values of the angles, which gives \(\sqrt{2}/2 - \sqrt{3}/2\).

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Most popular questions from this chapter

(a) use a graphing utility to graph the function and approximate the maximum and minimum points on the graph in the interval \([0,2 \pi),\) and (b) solve the trigonometric equation and demonstrate that its solutions are the \(x\) -coordinates of the maximum and minimum points of \(f .\) (Calculus is required to find the trigonometric equation.) Function $$f(x)=2 \sin x+\cos 2 x$$ Trigonometric Equation $$2 \cos x-4 \sin x \cos x=0$$

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Find all solutions of the equation in the interval \([0,2 \pi)\). $$\sin x-2=\cos x-2$$

Solve the multiple-angle equation. $$\sec 4 x=2$$

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