GSE Pre-Calculus TRIGONOMETRY CONCEPTS Intro to Trigonometry

Unit 1: Intro to Trigonometry

Trigonometry Concepts.

Trigonometry concepts are applied to many different applications other than triangles. The triangle relationships are still present, but for the most part the actual triangle is not drawn. While you are learning how to solve trigonometric problems, it is completely acceptable to draw diagrams with the right triangles to help you see the ratios. Line of Sight and Angles of Inclination

The line of sight is a straight line along which an observer observes an object. It is an imaginary line that stretches between observer's eye and the object that he is looking at. If the object being observed is above the horizontal, then the angle between the line of sight and the horizontal is called angle of elevation. If the object is below the horizontal, then the angle between the line of sight and the horizontal is called the angle of depression. Example 1

A Pre-Calculus student spots a drone hovering over the school football field. The field is approximately 80 yards away from where the student is standing. Using a simple string and protractor, the student is able to calculate the angle of elevation at 45 degrees. Using the basic trigonometry ratios, calculate the approximate height of this drone.

ANSWER: If we dropped a straight line from the drone to the ground, we can create a right triangle. The angle of elevation will be marked just above the horizontal line of sight. We want to use the tangent because it relates the ratio of the adjacent side (80 yards) to the opposite side (height). 𝑡𝑎𝑛𝜃 = 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒/𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑡𝑎𝑛45° = ℎ𝑒𝑖𝑔ℎ𝑡/80 80𝑡𝑎𝑛45° = ℎ𝑒𝑖𝑔ℎ𝑡 𝒉𝒆𝒊𝒈𝒉𝒕 = 𝟖𝟎(𝟏) = 𝟖𝟎𝒚𝒂𝒓𝒅𝒔

45°

80𝑦𝑎𝑟𝑑𝑠

GSE Pre-Calculus TRIGONOMETRY CONCEPTS Intro to Trigonometry

Unit 1: Intro to Trigonometry

Example 2

A flag pole casts a shadow 45 ft long. Find the height of the flag pole if the angle of elevation of the sun is 31.2°.

ANSWER: We can assume the flagpole is perpendicular to the ground and not leaning like the famous tower in Pisa. We can create a right triangle if we draw a line from the top of the shadow to the top of the flagpole. The angle of elevation will be marked just above the horizontal line of sight. We will want to use the tangent because it relates the ratio of the adjacent side to the opposite side.

𝑡𝑎𝑛𝜃 = 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒/𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 tan(31.2°) = ℎ𝑒𝑖𝑔ℎ𝑡/45 45tan(31.2°) = ℎ𝑒𝑖𝑔ℎ𝑡𝑜𝑓𝑓𝑙𝑎𝑔𝑝𝑜𝑙𝑒 At this point, we will need to use a calculator to help us find the tangent of an odd number like 31.2. Make sure you have a calculator with the trigonometric functions and know how to use the function. Also note: Values for trigonometry functions can be expressed in both degrees or radians. Make sure your calculator is reading the same type of units you are inputting into the function.

𝒉𝒆𝒊𝒈𝒉𝒕 = 𝟒𝟓(𝟎.𝟔𝟏°) = 𝟐𝟕.𝟐𝟓𝒇𝒆𝒆𝒕 DEGREES VS. RADIANS

When computing Trigonometry functions on a calculator, it is important to specifiy wheather you are measuring your angle in degrees or radians. This is similar to saying that it is some water is at 50o. This can mean drastically different things if the units are degrees Fahrenheit versus degrees Celsius. It might be confusing, but check out your calculaotor units before inputting your angle measure. To convert, remember there are 360 degrees in a full circle (a right angle is 90 degrees), and 2𝜋 radians in a full circle (there are 𝜋/2 radians in a right angle).

31.2°

45𝑓𝑒𝑒𝑡

GSE Pre-Calculus TRIGONOMETRY CONCEPTS Intro to Trigonometry

Unit 1: Intro to Trigonometry

Example 3

From a point on the ground 500 feet from the base of a building, an observer finds the angle of elevation to the top of the building is 24° and that the angle of elevation to the top of a cellular tower atop the building is 27°. Find the height of the building and the length of the cell tower.

ANSWER: We can create two right triangles if we draw lines from the ground to the top of the building and to the top of the cell tower. Again, we will use the tangent because it relates the ratio of the adjacent side to the opposite side.

𝑡𝑎𝑛𝜃 = 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒/𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 tan(24°) = ℎ𝑒𝑖𝑔ℎ𝑡/500

𝑡𝑎𝑛𝜃 = 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒/𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 tan(27°) = ℎ𝑒𝑖𝑔ℎ𝑡/500

500tan(24°) = ℎ𝑒𝑖𝑔ℎ𝑡𝑜𝑓𝑏𝑢𝑖𝑙𝑑𝑖𝑛𝑔 500 tan(27°) = ℎ𝑒𝑖𝑔ℎ𝑡𝑜𝑓𝑏𝑢𝑖𝑙𝑑𝑖𝑛𝑔𝑝𝑙𝑢𝑠𝑡𝑜𝑤𝑒𝑟 Solving for the heights gives us:

𝟐𝟐𝟐.𝟔𝟏𝒇𝒆𝒆𝒕 = 𝒉𝒆𝒊𝒈𝒉𝒕𝒐𝒇𝒃𝒖𝒊𝒍𝒅𝒊𝒏𝒈 𝟐𝟓𝟒.𝟕𝟔𝒇𝒆𝒆𝒕 = 𝒉𝒆𝒊𝒈𝒉𝒕𝒐𝒇𝒃𝒖𝒊𝒍𝒅𝒊𝒏𝒈𝒑𝒍𝒖𝒔𝒕𝒐𝒘𝒆𝒓

We have calculated our values for the various heights, but the question asks specifically for the height of the tower by itself. To find the height of the tower, we will need to subtract the height of the building from the height of the building plus tower to get the height of the tower by itself.

𝒉𝒆𝒊𝒈𝒉𝒕𝒐𝒇𝒕𝒐𝒘𝒆𝒓 = (𝟐𝟓𝟒.𝟕𝟔 − 𝟐𝟐𝟐.𝟔𝟏) = 𝟑𝟐.𝟏𝟓𝒇𝒆𝒆𝒕

27°

500𝑓𝑒𝑒𝑡

24°

GSE Pre-Calculus TRIGONOMETRY CONCEPTS Intro to Trigonometry

Unit 1: Intro to Trigonometry

Concept Check

1. From the top of a vertical cliff 40 m high, the angle of depression of an object that is level with the base of the cliff is . How far is the object from the base of the cliff? 2. A bird sits on top of a lamppost. The angle of depression from the bird to the feet of an observer standing away from the lamppost is . The distance from the bird to the observer is 25 meters. How tall is the lamppost? 3. To measure the height of the cloud cover at an airport, a worker shines a spotlight upward at an angle of 75° from the horizontal. An observer 600 m away measures the angle of elevation to the spot of light to be 35°. Find the height of the cloud cover.

34°

35°

34°

40𝑚𝑒𝑡𝑒𝑟𝑠

35°

25𝑚𝑒𝑡𝑒𝑟

𝑠