91Ó°ÊÓ

Use the composite argument properties to show that the given equation is an identity. $$\sec \left(\theta-90^{\circ}\right)=\csc \theta$$ (Be clever!)

The graph of each function is a sinusoid. a. Plot the graph of the given function. b. From the graph, find the equation of the sinusoid. c. Verify algebraically that the equation is sinusoidal. $$y=\cos x+\sin x$$

Exact Values Problems: a. Use the double and half argument properties to find the exact values of the functions, using radicals and fractions if necessary. b. Show that your answers are correct by finding the measure of \(A\) and then evaluating the functions directly. If \(\cos A=\frac{3}{5}\) and \(A \in\left(180^{\circ}, 270^{\circ}\right),\) find \(\sin 2 A\) and \(\cos \frac{1}{2} A\)

Solve the equation algebraically. Use the domain \(x \in[0,2 \pi]\) or \(\theta \in\left[0^{\circ}, 360^{\circ}\right]\) $$-8 \cos x-3 \sin x=5$$

Sine Double Argument Property Derivation Problem: Starting with \(\sin 2 x=\sin (x+x)\) derive the property \(\sin 2 x=2 \sin x \cos x\)

Musical Note Problem: The Nett sisters, Cora and Clara, are in a band. Each one is playing the note \(A\). Their fricnd Tom is standing at a place where the notes arrive exactly a quarter cycle out of phase. If \(x\) is time in seconds, the function equations of Cora's and Clara's notes are Cora: \(y=100 \cos 440 \pi x\) Clara: \(y=150 \sin 440 \pi x\) PICTURE CANT COPY a. The sound Tom hears is the sum of Cora's and Clara's sound waves. Write an equation for this sound as a single cosine with a phase displacement. b. The amplitudes 100 and 150 measure the loudness of the two notes Cora and Clara are playing. Is this statement true or false? "Tom hears a note 250 units loud, the sum of 100 and \(150 .\) " Explain how you reached your answer. c. The frequency of the A being played by Cora and Clara is 220 cycles per second. Explain how you can figure this out from the two equations. Is the following true or false? "The note Tom hears also has a frequency of 220 cycles per second."

Use the composite argument properties with exact values of functions of special angles (such as \(30^{\circ}, 45^{\circ}, 60^{\circ}\) ) to show that these numerical expressions are exact values of \(\sin 15^{\circ}\) and \(\cos 15^{\circ} .\) Confirm numerically that the values are correct. $$\cos 15^{\circ}=\frac{\sqrt{6}+\sqrt{2}}{4}$$

Identity Problems: Prove that the given equation is an identity. $$\tan \beta=\frac{1-\cos 2 \beta}{\sin 2 \beta}$$