Trigonometry

 
# 1
 Josh owns a triangular piece of land with two angles measuring 100o and 50o.The side opposite the 100o angle is 1.2km long.  Help Josh find the sizes of the other angle and the other sides.  To find the other sides use the Law of Sines. 
# 2
 Steve was riding his bike. The wheel has a radius of 10cm. The wheel has a central angle that intercepts an arc of 22cm. What is the size of the central angle of the bike in radians?
# 3 Find the area of this piece of land.


 
# 4  Anne rode on a ferris wheel. The graph models Anne's height (m) above the ground in relation to time (s). The data was recorded while the ride was in progress.
Questions:

a) What is the height of the axle on the ferris wheel? 

b) What is the radius of the ferris wheel? 
c) How long does it take the ferris wheel to complete one revolution? 
d) How fast is Anne traveling around the ferris wheel?            
e) How high above the ground is Anne when she boards the ferris wheel? 
f) Within the first 10s, how many times is Anne 9m above the ground?1m?                                                        g) What is the approximate height above the ground at 2.50s? 4.50s? 9.50s? 30.50s?

 
 
# 5 Find the radius of the following circle with designated arc = 22 cm, given side = 38cm, given side = 28cm, given Ð = 34°. 

     
# 6 Given 
1.State the transformations of 
2. State mapping diagram.
3. State key angles.
4. State equation of sinusoidal axis, phase shift, period, and amplitude.
# 7

 

 Two scuba divers are swimming. When they are 20m apart they see a shark directly below them. If the angle of depression from the first diver to the shark is 47° and the angle of depression from the second diver to the shark is 40° how far is the first diver from the shark? 
# 8
Dave has won the contract to replace the outhouses at Laurie Park, N.S. The front wall of each outhouse is 244cm high and the back wall of each outhouse is 214cm high, the side is 122 cm wide. The two sides of the outhouses are to be covered with aluminum panels.
A. Determine the area of aluminum panels needed
B. Find the angle that the rafter makes with the front wall.

 
# 9 Jeff wants to build a birdhouse for the bluebird that he received for his birthday. The diagram below shows the front of the birdhouse with specifications. The entrance for the bird is a circular hole with a diameter of 3.8cm. What is the area of wood to be painted for this side of the birdhouse? Note the area of a triangle is base multiplied by height, divided by 2.


 
# 10 Sharon is building the base for an octagonal gazebo. Each side of the octagon measures 1.3m. Determine the area of the floor of the gazebo so that Sharon knows how much wood to order. 

# 11 A diagonal of a rectangle makes a 38° angle with the longer side. If the longer side is 20, what is the length of the diagonal?
# 12 Wire stays are used to support a four-metre mast on sailboat. Rhonda needs to replace the stays that snapped in a storm. What is the length of each stay?

# 13 A kite is tied to a peg in the ground with a string 105 meters long. The string makes an angle of 42.6 degrees with the ground. How high is the kite? 
# 14 Bob is planning on raising a fence around his triangular shaped garden. He knows that side A is 5m, and that side B is 7m. Bob also knows that the two sides are joined together at  an angle of 56 degrees. How can he find the length of the unknown side?
# 15 While sitting atop a cliff (10 m directly above the shore) at the end of Blackett Street, George notices one of the cruise ships leaving Sydney harbor. He estimates an angle of depression of 0.15° to the ship. How far is the ship from the shore?
# 16 Mr. Nedal Awn has a triangular piece of property  that he would like to sod.  If sod costs $1.39 per square meter, what would it cost him?

# 17 If the large wheel shown below rotates 200º, how many degrees does the smaller wheel rotate? The radii are shown in each diagram.

# 19 A surveyor has taken measurements for the following property. He is not able to measure the fourth side due to a small pond and a wooded area. Determine the the length of the fourth side and the area of the property.

# 20 A search plane flying 2000m directly above the rescue ship spots the disabled trawler at an angle of depression of 16 degrees . How far away is the trawler from the rescue ship?
# 21 The school booster club is making a huge pennant to hang on the outside of the school. The triangular piece of material has two of its sides measuring 6m and 18m with the angle between them, 78 degrees. How much area do they have for their design?
# 22 While exploring the woods at the end of Bengal Road in Mira, two of Glace Bay High's finest, spotted a fire tower in the distance.  From where they were standing, they estimated an angle of elevation of 15° to the top of the tower.  Moving 40m closer to the tower, they now estimated the angle of elevation to be 20°.  How tall is the tower?
# 23 List all parts of the sinusoidal curve represented by the following function and use those parts to sketch the graph.

# 24 Evaluate the following without using the calculator:

# 25  As a recent graduate of Roller Coaster College and an expert on sinusoidal curves, your first job was designing the wave part of an exciting new roller coaster ride for downtown Glace Bay. Horizontally 10m into your wave part, you will arrive at the top of your sinusoidal wave. Another 50m horizontally will get you the bottom of your wave. The lowest point is 3m below ground level and the vertical distance between the highest and lowest points is 30m. Find an equation to model this part of the ride.
# 27 A triangular piece of property has two of its sides  measuring 140mand 250m. What is the area of this property if the angle between these sides is 15o?
# 28 Two of our Canadian Snow Birds are flying at the same altitude, one merely 200 meters in front of the other.  The one in front spots the landing strip at an angle of depression of  7 degrees.  The second plane at the same time sees the same landing strip at an angle of depression of 6.5 degrees.  How high above the ground are they flying?
# 29 List all the parts of a sinusoidal curve represented by the following function and use those parts to sketch its graph.

# 30 Evaluate the following without using the calculator:

# 31 Solve for all the values of x in degrees.

10=-13cos4(x-40°)+5

# 32 An old rock formation is warped in the shape of a sinusoidal.  Geologists studying that formation set up there instruments 20 meters to the left of the highest part of the rock which is about 10 meters above ground level.  They estimate that 60 meters to the right of the highest point and 30 meters below ground level is the lowest point.  Find the equation to model this rock formation.
# 33 Verify each of the following by using the trigonometric identities:

a) csc x tan x = secx                     b) (secx-1)(secx+1)(cos2x)=sin2x

# 35 Solve for all possible values of  x (in degrees)  sec x cot x = 3       
# 36  Due to poor landing conditions, a small plane is forced to circle Sydney airport for 25 minutes.  If the plane is traveling at 130km/h and remains 18km from the control tower, how many degrees has the plane rotated?  
# 37 A small pebble is stuck in Jim's bike tire.  The wheel has a radius of 34cm.  Determine the equation of the function that expresses the height of the pebble from the ground in terms of the distance that the bike travels from the point where the pebble got stuck in the tire.
# 38 Evaluate:  Sin 150º Cos 60º + Sin 60º Cos 330º without using the calculator.
# 39

A cosine graph has a period of 180, a vertical translation of +1, a vertical stretch of 2, and a horizontal translation of +90. Answer the following:  

1)What is the equation of the sinusoidal axis?         2) What is the amplitude?

3)What is the horizontal stretch?                               4) Where does the curve begin? (coordinates)

5)Where does the curve end? (coordinates)               6) How is the phase shift effected?

7)Write a mapping rule.                                                 8) Write in transformational form.

9)Draw the graph.  

 

# 40 Prove the following trig identity:    
# 41 A pole on a hill casts a shadow 55m down the hill.  The angle of depression of the shadow is 48 degrees while the angle of depression of the hill is 21 degrees.  Find the height of the pole.
# 42 Given triangle ABC. Angle C= 35 degrees; side a= 5.7 cm.; side c= 3.5 cm. Solve this triangle. 
# 43 A ship passing an island establishes by sonar a distance of the island of 3.5km from the ship to one end of the island and 5.1 km to the other end. The angle contained between the tip of the island is 115. Find the length of the island. 
# 44 Solve the following trig identity:    
# 45  Solve the following trig identity:     
# 46

 As you stop your car at a traffic light, a pebble becomes wedged between the tire treads. When you start off the distance of the pebble from the pavement varies sinusoidally with the distance you have traveled. If the diameter of the wheel is 24 inches. Draw the graph and state an equation for this situation. 

 
# 47

Determine the amplitude, period, and equation of the sinusoidal axis. State the mapping rule.