Exam 5: Analytic Trigonometry

Prove the identity: $\frac{\sin ^{3} A-\cos ^{3} A}{\sin ^{2} A-\cos ^{2} A}=\frac{2+\sin 2 A}{2(\sin A+\cos A)}$

Use a sum or difference identity to find the exact value of sin $165^{\circ} .$

For positive values of a,c and γ where $\gamma<180^{\circ}$ and c<a sin γ, how many triangles can be formed?

Solve the equation $2 \sin ^{2} x \cos x=\cos x$ by factoring and/or extracting square roots.

Prove the identity: $\frac{\sin \theta}{1+\cos \theta}+\frac{1+\cos \theta}{\sin \theta}=2 \csc \theta$

Solve the inequality $|\cot x| \sqrt{3} \text { for } 0 \leq x \leq 2 \pi$

Kelly must find the distance between points A and C on opposite sides of a lake. She locates a point B that is 425 ft from A and 672 ft from C . If the angle at B is $68^{\circ}$ what is the distance AC ?

Due to the wind, a tree grew so that it leaned $9^{\circ}$ from the vertical. From a point on the ground 30 meters from the base of the tree, the angle of elevation to the top of the tree is $26.3^{\circ}$ What is the height of the tree?

Use the half-angle identities to express cos 4C in terms of a function of 8C . 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.

Due to the wind, a tree grew so that it leaned $8^{o}$ from the vertical. From a point on the ground 28 meters from the base of the tree, the angle of elevation to the top of the tree is $24.6^{\circ}$ What is the height of the tree?

Which of the following is not a valid completion of the double-angle identity for $\cos 2 \theta ?$

Suppose the angle $\theta$ is formed by two sides of length 8 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 25 .

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

Use the fundamental identities to change the expression to one involving only sines and cosines. Then simplify. Show all your steps. $\tan ^{2} x-\frac{\csc ^{2} x}{\cot ^{2} x}$

What is the general solution to $\sin x \tan x-\sin x+\tan x-1=0 ?$

If a=9, b=8 , and $\alpha=65^{\circ}, \text { find } \beta \text {. }$

Confirm or disprove the following: 2 cos x=sin 2 x . If this is an identity, provide a confirmation. If it is not an identity, provide a counter example.

Factor the expression $\frac{\cot ^{2} x}{\csc x-1}$ and use fundamental identities to simplify.

The lengths of the sides of a triangular parcel of land are 200 ft, 250 ft , and 350 ft . What is the area, in square feet, of the parcel of land?

What is the area of $\triangle A B C$ if $\alpha=42^{\circ}$ b=17 , and c=25 ?