Tr igonometry Name: ___________________________________ Period: _________

Unit 5: Trigonometric and Periodic Functions Real World Applications Project

Part 1: You will create a collage of pictures illustrating all six trigonometric functions (sine, cosine, tangent, cosecant, secant, cotangent) found in nature (leaves, flowers, body parts, etc.), architecture (bridges, doorways, etc.), and everyday items (appliances, logos, furniture, etc.).  Requirements: Your project must contain: 1. Pictures of the entire objects where the trigonometric function is found 2. Different examples for each of the trigonometric functions - sine, cosine, tangent, cosecant, secant, cotangent (no repeat pictures are allowed) 3. Trace, in marker, the trigonometric function (with axis) in each picture 4. Title for the poster 5. CREATIVITY!!! Illustration: Your collage should be created using the following restrictions: * white or colored poster board * use scissors to cut out pictures (no tearing) * use glue to paste pictures (no taping) Grading: You will be graded according to the following  rubric:  

Category Points Possible Points

Example of Sine 1 point

Example of Cosine 1 point

Example of Tangent 1 point

Example of Cosecant 1 point

Example of Secant 1 point

Example of Cotangent 1 point

Examples of nature (at least 1) 1 point

Examples of architecture (at least 1) 1 point

Examples of everyday items 4 points (1pt for each)

Tracing of the trigonometric function (with axis) 6 points (1pt for each)

Title 2 points

Neat/Unique/Appropriate Up to 5 extra points

Tota l Po ints : ____ / 25

Part 2: 1. The tide, or depth of the ocean near the shore, changes throughout the day. The depth of the Bay of Fundy can be modeled by

𝑑 = 35− 28 cos𝜋6.2 𝑡

where d is the depth in feet and t is the time in hours . Consider a day in which 𝑡 = 0 represents 12:00 A.M. For that day, when do the high and low tides occur? At what time(s) is the water depth 3 !

! feet?

2. Cheyenne, Wyoming has a latitude of 41°N. At this latitude, the position of the sun can be modeled by

𝐷 = 31sin2𝜋365 𝑡 − 1.4

where t is the time in days and 𝑡 = 1 represents January 1. In this model, D represents the number of degrees due north or south of due east that the sun rises. Determine the days that the sun is more than 20° north of due east at sunrise.

3. A model for the average daily temperature T (in degrees Fahrenheit) in Kansas City, Missouri, is given by

𝑇 = 54+ 25.2 sin2𝜋12 𝑡 + 4.3

where t is measured in months and 𝑡 = 0 represents January 1. What months have average daily temperatures higher than 70℉? Do any months have average daily temperatures below 20℉? Solve the equation for the interval given. 4. 2 cos! 𝑥 − 3 cos 𝑥 = 0,                        0 ≤ 𝑥 < 2𝜋 5. sin! 𝑥 − sin 𝑥 = 2,                        0 ≤ 𝑥 < 2𝜋

Solve this equation for all values of x. 6. 3 tan! 𝑥 − 3 tan! 𝑥 − tan 𝑥 + 1 = 0 7. Each time your heart beats, your blood pressure first increases and then decreases as the heart rests between beats. The maximum and minimum blood pressures are called the systolic and diastolic pressures, respectively. Your blood pressure reading is written as systolic/diastolic. A reading of 120/80 is considered normal. A certain person’s blood pressure is modeled by the function

𝑝 𝑡 = 115+ 25 sin(160𝜋𝑡)

where 𝑝(𝑡) is the pressure in mmHg, at time t measured in minutes. (a) Find the period of p. (b) Find the number of heartbeats per minute.

(c) Graph the function p.

(d) Find the blood pressure reading. How does this compare to normal blood pressure? 8. In a predator/prey population model, the predator population is modeled by the function

𝑃 𝑡 = 900 cos 2𝑡 + 8000

where t is measured in years. (a) What is the maximum population? (b) Find the length of time between successive periods of maximum population.

Category Points Possible Points

Accuracy 16 (2 per question)

Work Shown 8 (1 per question)

Organization 1

TOTAL POINTS: __________ / 25