Algebra 2 Chapter 10 Note-Taking Guide Name _______________________ Trigonometry Per ____ Date ________________ 10.1A

10.1A Find each trigonometric ratio for θ :

10.1A Using Special Triangles to find the exact value of a trigonometric function.

10.1A Use trigonometric functions to find the value of x:

10.1A A builder is constructing a wheelchair ramp from the ground to a deck that is 18 inches high. The angle between the ground and the ramp must be 4.8°. To the nearest inch, what is the distance “d” between the end of the ramp and the deck?

10.1A A skateboard ramp will have a height of 12 inches and the angle between the ramp and the ground will be 17°. To the nearest inch, what is the length “x” of the ramp?

10.1A

A park ranger whose eye level is 5 feet above the ground measures the angle of elevation to the top of an eruption of Old Faithful geyser to be 34.6°. If the ranger is standing 200 feet from the geyser’s base, what is the height of the eruption to the nearest foot?

10.1A Work Space:

10.1B

10.1B Find the values of the six trigonometric functions for θ .

10.1B Find the values of the six trigonometric functions for θ .

10.1B Work Space

10.2 Angle Rotation Vocabulary: standard position: initial side: terminal side: positive rotation: negative rotation:

10.2 Draw angles with following measures... 300° -150° 900°

10.2 coterminal angles share the same terminal side when in standard position. Example: Find a positive and negative coterminal angle for each given angle.

(a) 40° (b) 88°

10.2 reference angle is the positive acute angle formed by the terminal side of the angle and the x-axis. example

(a) θ = 150° (b) θ = -130°

10.2 Find exact value of 6 trig ratios of angle whose terminal side goes through (4,-5).

10.2 Find the exact value of the six trig ratios given the point (-6,9) lies on the terminal side of the angle.

10.2 Work Space

10.3A What is a radian? It is a unit of angle measure based on arc length. An angle measures one radian if the length of the arc is the same as the radius of the circle. Since the circumference of the circle is 2 rπ , an angle representing one complete clockwise rotation measures 2π radians. We can use this fact to convert between radians and degrees.

10.3A The conversion formulas are

10.3A Convert each angle measure:

10.3A

10.3A Use the unit circle to find the exact value of each trigonometric function.

(a) cos210° (b) 5tan3π (c) sin315°

(d) tan180° (e) 4cos3π

(f) sinπ

10.3A

”All Students Take Calculus” helps us remember which trig function is positive in which quadrant. Use a reference angle to find the exact value of the sine, cosine, and tangent of each angle.

(a) 270° (b) -30° (c) 116π

Work space on next page…

10.3A Work Space

10.3B Developing the Unit Circle: Remember 45-45-90 triangle Remember 30-60-90 triangle

10.3B Developing the first quadrant of the unit circle: large version at the end of the packet!

Hand trick to work with reference angles: Calculate the following:

(a) sin6π

(b) cos6π

(c) sin3π

(d) cos3π

(e) sin4π

(f) cos4π

10.3B Working with tangent: Since tangent is really We can use the hand trick to calculate tangent by using bottom over top.

(a) tan6π

(b) tan3π

(c) tan4π

10.3B Extending beyond the first quadrant:

(a) 2cos3π

(b) 11sin

6π

(c) 3tan4π

(d) 5sin3π

(e) 7cos6π

(f) 3sin2π

(g) cosπ (h) tan 2π (i) cos2π

10.3B Work Space

10.4 We will use inverse trig functions to find angles.

Find all possible values:

(a) 1 2sin2

− (b) 1 3cos2

−

10.4

Remember that is this is a unit circle so the domain for sine and cosine is [-1,1]

10.4 Evaluate each inverse trigonometric function. Give your answer in both radians and degrees.

(a) 1 1Cos2

− (b) 1Sin 2−

10.4 Solve each equation to the nearest tenth. Use the given restrictions. (a) cos 0.6θ = , for 0 180θ° ≤ ≤ ° (b) cos 0.6θ = , for 270 360θ° ≤ ≤ °

10.4 A group of hikers plans to walk from a campground to a lake. The lake is 2 miles east and 0.5 mile north of the campground. To the nearest degree, in what direction should the hikers head?

10.4 A painter needs to lean a 30ft ladder against a wall. Safety guidelines recommend that the distance between the base of the ladder and the wall should be ¼ of the length of the ladder. To the nearest degree, what acute angle should the ladder make with the ground.

10.4 Work Space Continued on next page…

10.5A How do you find the area of the triangle?

What do you do when the height is unknown or hard to determine?

10.5A Area of any triangle (SAS): The area of any triangle can be found by half the product of two sides and the sine of the included angle.

10.5A Find the area of the triangle given a = 12, b = 15, and C = 57°

10.5A Law of Sines: The sine of any angle of a triangle over its opposite side is equal to the sine of another angle in the triangle over its opposite side. Law of Sines is best used in triangles where you know two angles OR two sides and a non-included angle (ASS).

sin sin sinA B Ca b c

= =

10.5A Solve the triangle. Round to the nearest tenth.

10.5A Work Space

10.5B Remember the Law of Sines:

10.5B When we have an ASS case, there is a possibility of having two triangles that meet that condition. • If the angle given is obtuse or right, solve like normal • If the angle given is acute, solve using the law of sines and check to see if the angle solved

for could be obtuse since there will be an acute and obtuse angle with that sine value.

(a) Maggie is designing a mosaic by using triangular tiles of different shapes. Determine the number of triangles that Maggie can form using the measurements a = 11cm, b = 17cm, and

30m A = ° . Then solve the triangles. Round to the nearest tenth.

10.5B Determine the number of triangular banners that can be formed using the measurements a = 50, b = 20, and 28m A = ° . Solve the triangle(s). Round to the nearest tenth.

10.5B Solve the following triangle: a = 8, b = 9, A = 35°

10.5B Work Space

10.6 Law of Cosines: It is used when you know all three sides (SSS) or two sides and the included angle (SAS).

2 2 2 2 cosc a b ab C= + − Or

( ) ( ) ( ) ( )( ) ( )2 2 2opposite side next to next to 2 next to next to cos included angle= + −

10.6 Solve the triangle. Round to the nearest tenth. (a) a = 8, b = 5, C = 32.2°

(b)

(c) a = 8, b = 9, c = 7

10.6 A coast guard patrol boat and a fishing boat leave a dock at the same time on the courses shown. The patrol boat travels at a speed of 12 nautical miles per hour (knots), and the fishing boat travels at a speed of 5 knots. After 3 hours, the fishing boat sends a distress signal picked up by the patrol boat. If the fishing boat does not drift, how long will it take the patrol boat to reach it at a speed of 12 knots?

10.6 AREA given SSS: Heron’s formula

If s is half the perimeter of a triangle 2

a b c+ +

, then the area of a triangle can be found by

( )( )( )Area s s a s b s c= − − −

(a) A blueprint shows a reception area that has a triangular floor with sides measuring 22ft, 30ft, and 34ft. What is the area of the floor to the nearest square foot?

10.6 (b) Find area of triangle given a = 12, b = 17, c = 15.

10.6 Work Space

Algebra 2 Trigonometry Application Notes Name: Section 1: In air and sea navigation, the angle measured clockwise from north to the line of travel is the

course, or heading of the plane or ship. The course for AC

is _______ and the course for AT

is ________.

Directions are also written by referring to North or South, using an acute angle. In the figure to the right, the direction of AC

= ___________ and AT

= ______________. Sample Problem 1: Two ships leave port at the same time. One ship travels on a course of 115̊ at 22 mi/hr, the other ship travels on a course of 207̊ at 18 mi/hr. How far apart are the two ships after 3 hours. Draw a picture and show your work. Section 2: In order to solve navigation problems, it is often necessary to interpret bearings. The clockwise angle from north to the line of sight to a point a point of reference is called the bearing of the point.

Sample Problem 2: The pilot of a plane traveling on a course of 30̊ sights the Anchorage International Airport. The pilot’s line of sight forms a right angle with the plane’s line of travel, as shown. Find the bearing of the airport.

Sample Problem 3: The pilot of a plane traveling on a course of 30̊ sights the Anchorage International Airport. The pilot’s line of sight forms a right angle with the plane’s line of travel, as shown. After traveling 65 miles on that course, the new bearing of the Anchorage International Airport is 238̊. Find the plane’s current distance from the Anchorage International Airport.

Section 3: Small errors in angle measurement can cause large errors in computed distances. To minimize such errors, surveyors measure angles using an extremely accurate instrument called a transit.

• When using angles accurately, we break angles down into minutes, and minutes down into seconds. So 1 degree = 60 minutes and one minute = 60 seconds. We label 20 degrees, 15 minutes, and 30 seconds as 20 15'30"° .

• If we would like to turn that into a decimal for calculation purpose, we would start with seconds and divide by 60. We would add that to the minutes. Then take the new minutes and divide by 60. Add that to the degrees and we are now in decimal form.

• To turn an angle into degrees, minutes, and seconds, take the decimal part only and multiply by 60. The whole number portion is the minutes, then take the decimal part and multiply by 60. The whole number portion is your seconds. Only use rounding rules on the seconds portion.

Sample Problem 4: Convert each measurement for A and B. Perform the operation for C and D

A.) 100.2125° B.) 80 7 '30"° C.) 32 42 '35"

51 07 '42"°

+ ° D.)

9048 07 '42"°

− °

Sample Problem 5: A bridge is being built across a river, from B to C. A surveyor using a transit determines that 43 20 'm A = ° . It is also known that the distance from A to C is 465 m. Find BC, the distance across the river.