Q: A park has a border of length $$180$$ yards, $$210$$ yards, and $$220$$ yards. What are the three angles formed at the intersection of the sides?

$$\alpha = 55^\\circ, \\beta = 65^\\circ, \\gamma = 60^\\circ$$

Question 1

Why must the sine of an angle always be less than one?

Accepted Answer

Because the sine function is defined as the ratio of the opposite side to the hypotenuse, and the hypotenuse is always the longest side in a right triangle.

Question 2

Solve for $$x in $$the right triangle where $$a = 10^2$, b = 20$, and c$$ = $$25^2.$$

Accepted Answer

$$x = 24$$

Question 3

Given $$c = 8$$, $$a = 6	$$heta^2$$, and $$	heta = $$0.32^2$$, solve for $$a.$$

Accepted Answer

$$a = 0.4096$$

Question 4

If $$k = 8x + b$$, $$b = 10x$$, and $$c = 12$$, solve for $$x.$$

Accepted Answer

$$x = 3$$

Question 5

Given $$y = 10^2$, b$$ = $$6x^2$$, and $$c = 5$$, solve for $$x.$$

Accepted Answer

$$x = 4$$

Question 6

A pilot flying a plane at an altitude of $$14,000$$ feet sees a lake at an angle of depression of $$32^\circ$$ and another at $$56^\circ. If $$the lakes are $$10$$ miles apart, find the distance between the lakes.

Accepted Answer

$$d = 12.4$$ miles

Question 7

A ship is flying directly above a town connecting two harbors. One harbor is seen at an angle of $$45^\circ$$ and the other at $$60^\circ. If $$the ship is $$1000$$ feet above the ground, what is the distance between the harbors?

Accepted Answer

$$d = 732 ft$$

Question 8

A park has a border of length $$180$$ yards, $$210$$ yards, and $$220$$ yards. What are the three angles formed at the intersection of the sides?

Accepted Answer

$$\alpha = 55^\circ, \beta = 65^\circ, \gamma = 60^\circ$$

Question 9

A regular heptagon has sides of length $$8$$ inches. Find the length of the diagonals of the heptagon.

Accepted Answer

$$d = 13.2 in$$

Question 10

If $$\sin(	heta) = 0.6$$ and $$	heta is in $$the second quadrant, what is $$\cos(	heta)$$?

Accepted Answer

$$\cos(	heta) = -0.8$$

Question 11

A building casts a shadow $$30$$ meters long when the angle of elevation of the sun is $$40^\circ. $$How tall is the building?

Accepted Answer

$$h = 19.3 m$$

Question 12

Given $$	an(\alpha) = 3/4$$ and $$\alpha is in $$the first quadrant, what is $$\sin(\alpha)$$?

Accepted Answer

$$\sin(\alpha) = 3/5$$

Question 13

If $$\cos(	heta) = -0.5$$ and $$	heta is in $$the third quadrant, what is $$\sin(	heta)$$?

Accepted Answer

$$\sin(	heta) = -\sqrt{3}/2$$

Question 14

A ladder $$20$$ feet long leans against a wall, making an angle of $$60^\circ$$ with the ground. How high up the wall does the ladder reach?

Accepted Answer

$$17.3 ft$$

Question 15

If $$\sin(2x) = 0.8$$, what is one possible value of $$x in$$ $$[0, \pi]$$?

Accepted Answer

$$x = 0.46$$

Precalculus Practice: Trigonometry, Angles, and Polygons

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