91Ó°ÊÓ

In Exercises 37 - 58, use the fundamental identities to simplify the expression. There is more than one correct form of each answer. \( \csc \phi \tan \phi + \sec \phi \)

In Exercises 65-68, use the co-function identities to evaluate the expression without using a calculator. \( \sin^2 25^\circ + \sin^2 65^\circ \)

The length of a shadow cast by a vertical gnomon (a device used to tell time) of height \( h \) when the angle of the sun above the horizon is \( \theta \) (see figure) can be modeled by the equation \( s = \dfrac{h \sin(90^\circ - \theta)}{\sin \theta} \). (a) Verify that the equation for \( s \) is equal to \( h \cot \theta \). (b) Use a graphing utility to complete the table. Let \( h = 5 \) feet. (c) Use your table from part (b) to determine the angles of the sun that result in the maximum and minimum lengths of the shadow. (d) Based on your results from part (c), what time of day do you think it is when the angle of the sun above the horizon is \( 90^\circ \)?

A weight is attached to a spring suspended vertically from a ceiling. When a driving force is applied to the system, the weight moves vertically from its equilibrium position, and this motion is modeled by \( y = \dfrac{1}{3} \sin 2t + \dfrac{1}{4} \cos 2t \) where \( y \) is the distance from equilibrium (in feet) and \( t \) is the time (in seconds). (a) Use the identity \( a \sin B \theta + b \cos B \theta = \sqrt{a^2 + b^2} \sin (B\theta + C) \) where \( C = \arctan (b/a) \), \( a > 0 \), to write the model in the form \( y = \sqrt{a^2 + b^2} \sin (Bt + C) \). (b) Find the amplitude of the oscillations of the weight. (c) Find the frequency of the oscillations of the weight.

A Ferris wheel is built such that the height \( h \) (in feet) above ground of a seat on the wheel at time \( t \) (in minutes) can be modeled by \( h(t) = 53 + 50 \sin \left(\dfrac{\pi}{16} t - \dfrac{\pi}{2}\right) \). The wheel makes one revolution every \( 32 \) seconds. The ride begins when \( t = 0 \). (a) During the first \( 32 \) seconds of the ride, when will a person on the Ferris wheel be \( 53 \) feet above ground? (b) When will a person be at the top of the Ferris wheel for the first time during the ride? If the ride lasts \( 160 \) seconds, how many times will a person be at the top of the ride, and at what times?

In Exercises 111 - 124, verify the identity. \( \left(\sin x + \cos x\right)^2 = 1 + \sin 2x \)

The range of a projectile fired at an angle \(\theta\) with the horizontal and with an initial velocity of \(v_{0}\) feet per second is \(r=\frac{1}{32} v_{0}^{2} \sin 2 \theta\) where \(r\) is measured in feet. An athlete throws a javelin at 75 feet per second. At what angle must the athlete throw the javelin so that the javelin travels 130 feet?

Consider the function given by \( f(x) = \sin^4 x + \cos^4 x \). (a) Use the power-reducing formulas to write the function in terms of cosine to the first power. (b) Determine another way of rewriting the function.Use a graphing utility to rule out incorrectly rewritten functions. (c) Add a trigonometric term to the function so that it becomes a perfect square trinomial. Rewrite the function as a perfect square trinomial minus the term that you added. Use a graphing utility to rule out incorrectly rewritten functions. (d) Rewrite the result of part (c) in terms of the sine of a double angle. Use a graphing utility to rule out incorrectly rewritten functions. (e) When you rewrite a trigonometric expression,the result may not be the same as a friends. Does this mean that one of you is wrong? Explain.