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Section 1.1: Angles Complementary and Supplementary Angles

Complementary angles have measures that add up to 90. Supplementary angles have measures that add up to 180.

Fractions of a Degree

One minute of angle is one-sixtieth of a degree:

One second of angle is one-sixtieth of a minute:

Example of an angle measurement stated in degrees, minutes, and seconds (DMS):

Converting from degrees, minutes, and seconds (DMS) to decimal degrees (DD) Example: convert to DD Divide the number of second by sixty to convert it to an equivalent number in minutes.

o

Add that to the number of minutes, then divide by sixty again to convert to an equivalent number in degrees.

o

Add that number to the number of degrees, and round to an appropriate number of places:o

Note that . Thus, stating DMS measurements to the nearest

thousandth of a degree won’t result in much round-off error, but stating your answer beyond a ten-thousandth of a degree implies a precision not conveyed by mere seconds.

Note: on a graphing calculator, you can type in a DMS measurement using the degree and minute symbols found in the ANGLE menu, and the quotation marks for the seconds.

Converting from decimal degrees (DD) to degrees, minutes, and seconds (DMS) Example: convert to DMS Multiply the decimal portion by sixty to convert it to an equivalent number in minutes.

o

Multiply the resulting decimal portion by sixty again to convert to an equivalent number in seconds.

o

Round to an appropriate number of places:o

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Note: on a graphing calculator, you can type in a DD measurement and convert it to a DMS measurement using the DMS function found in the ANGLE menu.

Standard Position An angle in standard position has its vertex at the origin of a Cartesian coordinate system

and its initial side on the positive x-axis. Thus, acute angles have their terminal sides in Quadrant I. Also, obtuse angles have their terminal sides in Quadrant II. A quadrantal angle has its terminal side on the x- or y-axis. Quadrantal angles have measure of 90, 180, 270, 360, 450, 540, 630, 720, …, -90, -

180, -270, -360,-450, etc.

Coterminal Angles Coterminal angles have the same terminal side, but differ in how many rotations around

the circle were taken to get to that side. The measures of coterminal angles differ by a multiple of 360. Angles coterminal with a given angle measure of will have measures given by

+ (360)n for any integer n.

Section 1.2: Angle Relationships and Similar Triangles

Geometric Properties of AnglesVertical angles are on opposite sides of the intersection point of two lines.

Vertical angels have equal measures.Parallel lines lie in the same plane and do not intersect.A transversal is a line that intersects two parallel lines.

Different angles formed by the transversal are given special names to reflect special relationships between their measures:

o Corresponding angles have equal measures (this is a postulate (i.e., a fact that is accepted without proof) which is used to prove the following relationships).

o Alternate interior angles have equal measures.o Alternate exterior angles have equal measures.o Interior angles on the same side of the transversal are supplementary (add to

180).

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TrianglesThe sum of the measures of the angles of any triangle is 180.

Types of TrianglesTriangles are classified according to their angles and sides.

Angle classifications:o Acute triangles have all angles less than 90.o Right triangles have one angle of 90.o Obtuse triangles have one angle greater than 90.

Side classifications:o Equilateral triangles have all sides the same length.o Isosceles triangles have two sides of the same length.o Scalene triangles have no sides of the same length.

Congruent triangles have all the same side lengths and all the same angles. Congruent is the proper way to say that two triangles are “equal” or “the same”.Similar triangles have the same shape (i.e., all the same angles), but not necessarily the same size. The larger triangle is like a “magnification” of the smaller one.

Conditions for Similar Triangles:If triangle ABC is similar to triangle DEF, then the following conditions must hold:

Corresponding angles must have the same measure (a postulate). Corresponding sides must be proportional.

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Section 1.3: Trigonometric Functions

The Six Trigonometric Functions are defined as follows:

(sine) (cosine)

(tangent) (cotangent)

(secant) (cosecant)

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Section 1.4: Using the Definitions of the Trigonometric Functions

The Reciprocal Identities (Section 1.4)

Note: we can re-state these reciprocal identities as:

Signs and Ranges of the Trigonometric FunctionsCombining the signs of x and y in the four quadrants with the definitions of the trig functions allows us to state the sign of the trig functions for any angle in a given quadrant.

Range of the Six Trigonometric FunctionsTrig Function Range stated in interval

notationRange stated in set builder notation

The Pythagorean Identities (Section 1.4)

The Quotient Identities (Section 1.4)

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Section 2.1: Trigonometric Functions of Acute Angles

Right Triangle Based Definitions of Trigonometric Functions (Section 2.1)(SOH CAH TOA)

Cofunction Identities (Section 2.1)For any acute angle A,

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Trigonometric Function Values for 45 (Section 2.1)

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Trigonometric Function Values for 30 and 60 (Section 2.1)

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Section 2.2: Trigonometric Functions of Non-Acute AnglesReference AnglesA reference angle is the angle formed between the terminal side of any angle and the x axis.Notice that the reference angle is always an acute angle. This means that the trig function values of acute angles can be used to compute the trig function values of any angle. All you have to do is get the signs right for x and y based on the quadrant of the terminal side.

Reference angle for terminal side in Quadrant II

Reference angle for terminal side in Quadrant III

Reference angle for terminal side in Quadrant IV

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Section 2.3: Finding Trigonometric Function Values Using a CalculatorEvaluating Sine, Cosine, and Tangent on a Calculator

Make sure your calculator is in degree mode.

Enter the trig function followed by the angle in parentheses.

For angles in DMS, you can enter the angle using the DMS functionality of your calculator.

For angles in DMS, you also can enter the angle as the number of degrees plus fractions of a degree.

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Evaluating Secant, Cosecant, and Cotangent on a Calculator Calculators do not have buttons for these functions. You have to use the reciprocal identities to evaluate these trig functions on a calculator.

The Reciprocal Identities (Section 1.4)

To enter the reciprocal calculations correctly, you have to enter 1 divided by the correct trig function. DO NOT use the SIN-1, COS-1, or TAN-1 buttons for cosecant, secant, or cotangent; those are the inverse functions (not the reciprocals).

Finding Angle Measures on a Calculator We can use this inverse function property notion to solve equations where a trig function

of an unknown angle is equal to a constant. We then just take the inverse function of both sides of the equation to find the angle.o The inverse sine function is written as sin-1.o The inverse cosine function is written as cos-1.o The inverse tangent function is written as tan-1.

Note that when you use the inverse trig functions on a calculator, o The sin-1 function returns an angle in the interval .o The cos-1 function returns an angle in the interval .o The tan-1 function returns an angle in the interval .

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To find an angle for the secant, cosecant, and cotangent functions, you have to use the reciprocal identities first to convert the equation so that it has cosine, sine, or tangent.

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Section 2.4: Solving Right Triangles The way we write a measurement can be used to indicate how precise the measurement was.

o If we write 15 inches, that means the object we measured was between 14.5 and 15.5 inches long.

o If we write 15.0 inches, that means the object we measured was between 14.95 and 15.05 inches long.

o If we write 15.00 inches, that means the object we measured was between 14.995 and 15.005 inches long.

o In other words, a measurement is good to ½ of the least significant digit.This is called using significant digits to write a number.When using trig functions, the rules for writing significant digits are as follows:

Angles and Accuracy of Trig FunctionsAngle Precision Trig Function Accuracy Example

1 2 sig. figs. , 0.1 or 10 3 sig. figs. ,

0.01 or 1 4 sig. figs. ,

0.001 or 10 5 sig. figs. ,

To solve a right triangle:1. Make a sketch of the triangle, label sides and angles consistently (a, b, and c for the legs

and hypotenuse; A and B for the complementary angles), and label the given information.2. Find a way to relate the unknown parts to the given information using:

a. a trig function (sine, cosine, or tangent), b. the Pythagorean Theorem (a2 + b2 = c2), c. or complementary angles. d. Try to use original given information to minimize rounding errors.

3. Check your work:a. Make sure the sides obey the Pythagorean Theorem.b. Make sure the angles add up to 180.c. Make sure unused trig functions give the right answers.d. Make sure that the longest side is opposite the largest angle, and the shortest side

is opposite the smallest angle.

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Section 2.5: Further Applications of Right Triangles

To solve applied right triangle problems:1. Make a sketch of the situation.2. Identify/draw right triangles on your sketch that connect given information to unknown

information.3. Solve the right triangle or triangles.

Bearing:In navigation, the word bearing means one of two things:

An angle measured clockwise from due north. An angle measured from either due north or due south in either a clockwise or

counterclockwise direction. The direction is specified as a rotation either to the east or the west. The starting direction is stated first, then the angle, then the direction of rotation of that angle from the starting direction.