Exam 5: Analytic Trigonometry

Use a grapher to conjecture whether the equation is likely to be an identity. If it is not, give a counter example. $\sin ^{2} x \cos ^{4} x=\cos ^{2} x+\cos ^{4} x-\cos ^{6} x$

Which of the following is correct?

If a=9, b=8 , and $\alpha=75^{\circ}, \text { find } \beta$

Suppose the angle $\theta$ is formed by two sides of length 10 in an isosceles triangle. Find an expression for the area $A(\theta)$ of the triangle. Then find the value(s) of $\theta$ so that the triangle's area is 37 .

A civil engineer needs to measure the length of a proposed tunnel through a mountain. The engineer chooses a point that can be seen from each end of the tunnel and measures the distance to each end of the tunnel to be 3,000 meters and 4,200 meters, respectively. If the angle between the two distances is $113^{\circ}$ what is the length of the tunnel?

Use the half-angle identities to express sin 3 C in terms of a function of 6C . In Problems 8-9, consider the triangle ABC . Express your answers as decimals rounded to the nearest hundredth. If more than one triangle is possible, give all possible answers.

Two points are 200 ft apart on opposite sides of a tree. The angles of elevation from each point to the top of the tree are $30^{\circ}$ and $35^{\circ}$ What is the height of the tree?

The angles of elevation to the top of a tower from two points that are on the same side of the tower and 30 meters apart are $37.6^{\circ} \text { and } 46.3^{\circ}$ What is the height of the tower?

In triangle ABC, $\alpha=42^{\circ}, \beta=25^{\circ}$ and b=6. Find the remaining sides and angle.

The lengths of the sides of a triangular parcel of land are 300 ft, 350 ft , and 450 ft . What is the area, in square feet, of the parcel of land?