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Pre-calculus Chapter 4 Part 1
NAME: ______________________________________________ P. _____
Date Day Lesson Assigned Due
2/12 Tuesday 4.3
Pg. 284: Vocab: 1-3. Ex: 1, 2, 7-13, 27-32, 43, 44, 47 a-c, 57,
58, 63-66 (degrees only), 69, 72, 74, 75, 78, 79, 81, 82, 86, 90,
94, 98.
2/14 Thursday 4.1 & 4.4
Pg. 265: Vocab: 3-4. Ex: 23-28, 30-32, 61-67, 102, 114, 118.
Pg. 294: Vocab: 1-7. Ex: 1, 4, 7, 10, 13-16, 18, 20, 37-40, 46,
48, 53-55, 65, 66, 72, 77-83, 111.
2/19 Tuesday 4.1 & 4.2
Pg. 265: Vocab: 4-8. Ex: 3-9, 11, 12, 18, 29, 34, 39-41, 43-45,
47, 48, 56, 57, 72, 74, 76, 82, 84, 90, 92.
Pg. 274: Vocab: 1-3. Ex: 18, 20, 24, 29, 30, 54, 56, 58, 90.
4.3
2/21 Thursday Review Review Worksheet
Extra Credit: Pg. 344: 4-36 by 4’s, 42-70 by 4’s, 82, 88, 94, 100.
4.1 & 4.4
2/25 Monday Test Test
4.1 & 4.2 Review Notes
Extra Credit
Notes:
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Chapter 4: Trigonometric Functions
4.3 Right Triangle Trigonometry
Basic Right Triangle Trigonometry
The basic right triangle trigonometric ratios are given by SOH-CAH-TOA
For reciprocal functions, you must turn the primary trig ratios upside down.
So,
Example 1: Use the right triangle below to find the exact values (trig ratios) of the six trigonometric functions of .
There are two types of special triangles for which you should memorize side lengths to help you build trig ratios.
These are 45-45-90 and 30-60-90 triangles.
Example 2: Fill in the sides of the triangles below. Then find the trig ratios for each acute angle.
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It is possible to find exact values of trig ratios when using special triangles or right triangles with two given sides.
However, you may need to find approximate values of trig ratios to solve real-world problems. Your calculator
will find approximate trig ratios in decimal form.
Make sure your calculator is set in the correct mode (degree or radian).
Example 3: Find the following trig ratios using your calculator.
a.
b. c.
d.
(Remember that
e. f.
A major application of trigonometry is solving for some (or all) of the missing parts of a right triangle. This
process is called solving right triangles. In addition to using trigonometric ratios, you will have to use some
geometry to solve right triangles. On common method is the Pythagorean Theorem.
Example 4: Find the missing part labeled for each triangle below.
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Example 5: Find the values of the six trigonometric functions of θ in the right triangle shown.
sin θ = csc θ =
cos θ = sec θ =
tan θ = cot θ =
Example 6: Sketch a right triangle corresponding to the trigonometric function of the acute angle θ. Use the
Pythagorean Theorem to determine the third side and then find the other 5 trigonometric functions of θ.
a. tan θ = 3
Example 7: A 30-meter line is used to tether a helium filled balloon. Because of a breeze, the line makes an
angle of approximately 75° with the ground.
a. Draw a right triangle that gives a visual representation of the problem. Label the known parts and use a
variable for the unknown parts.
b. What is the height of the balloon? (Use a trigonometric function to write an equation involving the
unknown height.)
9
15 θ
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Angle of Elevation: The angle measured from a
horizontal upward toward an object. (angle going up)
Angle of Depression: The angle measured from a
horizontal downward toward an object. (angle going
down)
Example 8: A surveyor standing 100 feet from the base of a large tree measures the angle of elevation to the
top of the tree to be 53.5. How tall is the tree?
Example 9: A sonar operator on a ship detects a submarine at a distance of 500 meters and an angle of
depression of 40. How deep is the submarine?
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4.1 & 4.4 Degree Measure and Trigonometric Functions of any Angle
Angles
An angle is formed by rotating a ray about its endpoint.
The starting point of the ray is the ____________ side.
The final position of the ray is the ____________ side.
The point of rotation is called the ____________ of the angle.
The amount and direction of rotation is the ____________ of the angle.
A (1) degree is
of a complete rotation.
If the rotation is counter-clockwise (CCW), the measure of the angle is _____________.
If the rotation is clockwise (CW), the measure of the angle is _____________.
Angles are named with Greek letters (α, β, θ) or Capital letters (A, B, C).
An angle in standard position is said to lie in the quadrant that contains its terminal side
An angle in standard position where its terminal side lies on the x or y- axis is called a quadrantal angle
Angles between 0° and 90° ( ) are acute
Angles between 90° and 180° ( )are obtuse
With calculators it is convenient to use decimal degrees to denote fractional parts of degrees. Historically,
however, fractional parts of degrees were expressed in minutes and seconds (‘ for minutes, “ for seconds).
1‘ = one minute =
( 1“ = one second =
(
64 degrees, 32 minutes and 47 seconds can be written as 64°32’47”
Example 1:
a. Convert 64°32’47” to decimal degree form
By hand:
b. Convert 152°15’29” to decimal degree form
By Calculator:
By Calculator: (2nd APPS for angle and min.)(Alpha + for sec.)
Example 2:
Convert 30.023° to DMS
(In the angle menu)
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Reference Angles Let θ be an angle in standard position. Its reference angle is the acute angle θ’ formed by the
terminal side of θ and the horizontal (x) axis. θ’ is always positive.
Example 3: Sketch angles in standard position having the following measures, and find their reference angles.
a. 315
b. -150 c. 420
A reference angle can be used to form a reference triangle whose trig ratios are the same as those for the
actual angle. A reference triangle is formed by drawing a vertical segment from the terminal side of the angle
to the x-axis, forming a right triangle. For a reference triangle, the hypotenuse is always considered positive. A
leg is considered to be positive if it is to the right of or above the origin. It is considered negative if it is to the left
or below the origin.
Example 4: Use the reference angles from Example 3 to build reference triangles, then find the following trig
ratios. Be careful with positive/negative signs!
a. sin 315
cos 315
tan 315
b. sin (-150)
cos (-150)
tan (-150)
c. sin 420
cos 420
tan 420
Example 5: Let (-3, 4) be a point on the terminal side of θ. Find the sine, cosine, tangent, cosecant, secant and
cotangent of θ. First draw θ with terminal side on (-3, 4).
Example 6: Find the values of all six trig functions of if and .
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If an angle does not have a reference angle of 30, 45, or 60 you can still find its trig ratios (in decimal form) by
using a calculator.
Example 7: Use a calculator to find the following:
The signs of trigonometric function values in the four quadrants can be determined easily from the definitions of
the functions. For instance, because
, cos θ is positive wherever x > 0, which is in quadrant I and IV.
(Remember, r is always positive.)
S A
T C
If angles in standard position share the same terminal side, they are called coterminal angles. The angles in the
diagram below have measures of 30, 390, and -330. They are coterminal.
Coterminal angles have measures which differ by multiples of 360. Thus, you can find angles which are
coterminal to a given angle by adding or subtracting 360 as many times as you wish.
Finding Reference Angles:
Example 8: Draw θ and then find the reference angle θ’ for each of the following. Also, find one negative and
one positive coterminal angle.
a. θ = 300° b. θ = c. θ = 405
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-1 1
-1
1
4.1 & 4.2 Quadrant Angles and Radian Measure for Angles
Suppose an angle in standard position has its reference triangle
labeled with x, y and r as shown in the figure.
Then using SOH-CAH-TOA:
and
Now, suppose you draw a circle with radius 1, so that it
circumscribes the reference triangle as shown. This circle is called
a unit circle, and it is useful in developing the trig ratios for the
quadrantal angles (…-270, -180°, -90°, 0°, 90°, 180°, 270°,…).
In the unit circle, r=1 so and
Example 1: Use the unit circle shown to do the following:
a. Plot points at every intersection of the unit circle with an axis.
b. Label these points using ordered pairs.
c. Fill in the following table. (Remember for the unit circle,
and
)
-90 0 90 180 270 360
sin
cos
tan
Quadrant angles do not form reference angles, so it is important to think of a unit circle when finding their trig
ratios without a calculator. For any other special angles (multiples of 30, 45 or 60you should build reference
triangles.
Example 2: Find the following trig ratios:
a. tan 720
b. csc (-270) c. sec 450
d. cos 540 e. cot (-90) f. sin 210
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In Geometry, you probably always measured angles in degrees. Many applications in trigonometry use angles
measured in degrees. However, upper level mathematics courses almost always use angles which are
measure in radians. One radian is defined to be the measure of the central angle (of a circle) that intercepts
an arc s which is equal in length to the radius of the circle.
Since the circumference of a circle is 2 r, it takes 2 arc
lengths of s=r to go all the way around the circle. Thus,
360=2 radians. One radian =
57.2958. Dividing
both sides of both sides of 360= 2 by 2, you get a very
convenient equation for converting angle measurement
types:
Why Radian's not degrees Online Radian Explanation
Since a circle’s circumference = 2πr, a full circle’s angle measure
is 2π radians. (Just over 6 radius lengths, 6.283 radians)
WHY are these the conversion equations?
Example 3: Convert the following, and tell which quadrant it lies in.
a. 135°
b. -270 °
c.
d. 2 rads
Angles between 0 and
(
) are acute
Angles between
and π (
)are obtuse
Note: when no units of angle measure are specified, radian measure is implied.
Some common angles measured in radians:
Conversion between Degrees and Radians
1. To convert Degrees to Radians
Multiply degrees by
2. To convert Radians to Degrees
Multiply radians by
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Some angles occur so often in trigonometry that you should try to memorize the degree-radian equivalencies.
They are fairly easy to develop if you do forget them.
Example 4: Find the following degree-radian equivalencies. Remember that 180= radians.
a. 90= b. 30 = c. 60 = d. 45 =
Example 5: Sketch each angle in standard position, and find the trig ratios asked for.
a.
sin
tan=
b.
cos
tan=
When finding coterminal angles for angles expressed in radian measure, you can add or subtract 2 as many
times as you need to.
Example 6: Sketch each angle in standard position, and find one positive and one negative coterminal angle
for each. (Express your answers in radian measure)
a.
b.
Your calculator will easily find trig ratios for any angle measured in either degrees or radians. However, you
must let the calculator know which type of measure that you are using. The two measures (radian and degree)
are both found in the calculator’s MODE.
Example 7: Use your calculator to find the following to 3 decimal place accuracy.
Note: When degrees are not specifically indicated, angle measures are considered to be in radians.
a. cos 3.725 b. tan(
) c. csc 2.621
Arc Length
For a circle of radius r, a central angle θ intercepts an arc of length
s given by s=r θ
Where θ is measured in radians. If r = 1 then s = θ, and the radian
measure of θ equals the arc length.
Example 8: If a clock hand moves from 12:00 noon to 12:20 pm and the minute hand is 6 inches long, what is
the length of the arc it sweeps out?
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Example 9: A circle has a radius of 4 inches. Find the length of the arc intercepted by a central angle of 240°.
Example 10: Glasgow, MT is due north of Albuquerque, NM. Find the distance between Glasgow (48°9’) and
Albuquerque (35°5’). Assume the radius of the earth is 3960 miles.
Even and Odd Trig Functions The cosine and secant functions are even.
cos(-t) = cos t sec (-t) = sec t
The sine, cosecant, tangent and cotangent functions are odd.
sin(-t) = - sin t csc (-t) = - csc t
tan(-t) = - tan t cot (-t) = - cot t
Example 11: Given cos t = -3/4 find the following:
a. (
b. (
Example 12: Evaluate the following
a. (
b. (
c. (
d. (