Geometry B, Unit 10 (All lessons)

inclinometer

a device for measuring the amount of incline or tilt of an object or a surface

The hot air balloon is 500 feet off the ground. The observer sees his landing zone at an angle of depression of 45°. Find the horizontal distance to his landing spot.

1a. d 2. j 3. b 4. e 5. f

Find the length of c . Use the values sin30∘=0.5,cos30∘=0.866,tan30∘=0.577

1a. e 1b. i 2. a 3. l 4. g 5. j

Find the measure of ∠A .

1a. e 1b. i 2. j 3. d 4. f

Find the length of b . Use the values sin30∘=0.5,cos30∘=0.866,tan30∘=0.577 .

1a. h 1b. a 2. j 3. l 4.g 5. f

Given the angle of elevation and distance from the Eiffel tower, find its height (x) .

1a. k 1b. j 2. d 3. i 4. b 5. f

If h=6 in; y=18 in then find x.

2 in

Find the length of side b .

2.85 cm

Prove: ΔABC~ΔCDB~ΔADB

3. Def. of Altitude 5. Reflexive 6. AA 7. Reflexive 8. AA 9.Transitive

Find the length of a .

3.7 cm

Find tan D.

3/4

Find cos F.

3/5

Find sin D.

3/5

If c=10 cm; x=1.6 cm; find s1 .

4 cm

If x=2 cm; y=8 cm then find the length of the altitude h.

4 cm

Find tan F.

4/3

Find cos D.

4/5

Find sin F.

4/5

Find m∠B to the nearest degree.

47°

If c=10 cm; y=8.4 cm; find s2 .

9.17 cm

Mnemonic

A mnemonic device that is commonly used to help students remember the primary trigonometric functions is SOHCAHTOA, pronounced [soh-kuh-TOH-uh]. It stands for: SOH = Sine - Opposite over Hypotenuse CAH = Cosine - Adjacent over Hypotenuse TOA = Tangent - Opposite over Adjacent Or, you could use this one: Some Old Horses Can't Always Hide Their Old Age.

Find the arithmetic mean and geometric mean of the set of numbers.{2, 18}

Arithmetic mean: 10 Geometric mean: 6

Find the arithmetic mean and geometric mean of the set of numbers. {2, 32}

Arithmetic mean: 17 Geometric mean: 8

The length of the altitude to the hypotenuse of a right triangle is the geometric mean of the lengths of the two segments of the hypotenuse.

Cor. 1 of Thrm. 10.1

The length of a leg of a right triangle is the geometric mean of the lengths of the hypotenuse and the segment of the hypotenuse adjacent to that leg.

Cor. 2 of Thrm. 10.1

___ is a trigonometric ratio consisting of the length of the hypotenuse to the length of the side opposite a given angle of a right triangle; the reciprocal of the sine.

Cosecant

___ is a trigonometric ratio consisting of the length of the side adjacent to a given acute angle of a right triangle to the length of the hypotenuse.

Cosine

___ is a trigonometric ratio consisting of the length of the side adjacent to a given acute angle of a right triangle to the length of side opposite the given angle; the reciprocal of the tangent.

Cotangent

___ is a trigonometric ratio consisting of the length of the side opposite a given acute angle of a right triangle to the length of the side adjacent to the given angle.

Tangent

Greek mathematician and teacher who lived in the 2nd century B.C.

Hipparchus

Which statement is NOT correct?

If you know the measure of all three angles of a right triangle, you can find the length of each of the three sides.

4) Choose the law and the formula that would be used to solve the triangle.

Law: Law of Cosines Formula: a^2 = b^2 + c^2 - 2bcCOS A

5) Choose the law and the formula that would be used to solve the triangle.

Law: Law of Cosines Formula: b^2 = a^2 + c^2 - 2acCOS B

6) Choose the law and the formula that would be used to solve the triangle.

Law: Law of Cosines Formula: c^2 = a^2 + b^2 - 2abCOS C

3) Choose the law and the formula that would be used to solve the triangle.

Law: Law of Sines Formula: sinA / a = sinB / b

2) Choose the law and the formula that would be used to solve the triangle.

Law: Law of Sines Formula: sinA / a = sinC / c

1) Choose the law and the formula that would be used to solve the triangle.

Law: Law of Sines Formula: sinB / b = sinC / c

___ is a trigonometric ratio consisting of the length of the hypotenuse to the length of the side adjacent to a given angle of a right triangle; the reciprocal of the cosine.

Secant

Theorem 10.1

The altitude to the hypotenuse of a right triangle forms two triangles that are similar to each other and to the original triangle.

Corollary 2 of Theorem 10.1

The length of a leg of a right triangle is the geometric mean of the lengths of the hypotenuse and the segment of the hypotenuse adjacent to that leg.

Corollary 1 of Theorem 10.1

The length of the altitude to the hypotenuse of a right triangle is the geometric mean of the lengths of the two segments of the hypotenuse.

The altitude to the hypotenuse of a right triangle forms two triangles that are similar to each other and to the original triangle.

Thrm. 10.1

___ ratios are the ratios of the lengths of the two sides of a right triangle.

Trigonometric

unit circle

a circle with a radius of one unit that has its center at the origin on the coordinate plane

secant

a trigonometric ratio consisting of the length of the hypotenuse to the length of the side adjacent to a given angle of a right triangle; the reciprocal of the cosine

cosecant

a trigonometric ratio consisting of the length of the hypotenuse to the length of the side opposite a given angle of a right triangle; the reciprocal of the sine

cotangent

a trigonometric ratio consisting of the length of the side adjacent to a given acute angle of a right triangle to the length of side opposite the given angle; the reciprocal of the tangent

cosine

a trigonometric ratio consisting of the length of the side adjacent to a given acute angle of a right triangle to the length of the hypotenuse

sine

a trigonometric ratio consisting of the length of the side opposite a given acute angle of a right triangle to the length of the hypotenuse

tangent

a trigonometric ratio consisting of the length of the side opposite a given acute angle of a right triangle to the length of the side adjacent to the given angle

radian

a unit of angular measure equal to the length of the arc divided by the radius of the arc

Match the trigonometric name with the correct ratio. cos

adjacent / hypotenuse

angle in standard position

an angle with its vertex at the origin and one of its rays on the x-axis of the coordinate plane

useful for finding the average of a set of values that are similar

arithmetic mean

solving a triangle

calculating unknown side lengths or angle measures of a triangle if certain of the side lengths and/or angle measures are known

Angle of ___ is the angle formed by a horizontal line and a line of sight to a point below the horizon.

depression

Angle of ___ is the angle formed by a horizontal line and line of sight to a point above the horizon.

elevation

Law of Cosines

for any ABC with side lengths a, b, and c: a^2 = b^2 + c^2 - 2bcCOS A b^2 = a^2 + c^2 - 2acCOS B c^2 = a^2 + b^2 - 2abCOS C

Law of Sines

for any ABC with side lengths a, b, and c: sin A/a = sin B/b = sin C/c

The ___ is the positive nth root of the product of n factors.

geometric mean

terminal side of an angle

he ray of an angle in standard position that does not lie on the x-axis

A(n) ___ is a device for measuring the amount of incline or tilt of an object or a surface.

inclinometer

The three expressions, sin-1, cos-1, and tan-1 are called _____ trig functions and are used to find the measure of the acute angles of a right triangle if you know the lengths of at least two sides.

inverse

reciprocal

one of a pair of values whose product is one; also called the multiplicative inverse

Match the trigonometric name with the correct ratio. tan

opposite / adjacent

Match the trigonometric name with the correct ratio. sin

opposite / hypotenuse

One of a pair of values whose product is one is called a(n) ___; also called the multiplicative inverse.

reciprocal

The ___ is a trigonometric ratio consisting of the length of the side opposite a given acute angle of a right triangle to the length of the hypotenuse.

sine

The process known as _____ a triangle is used for calculating unknown side lengths or angle measures of a triangle if certain of the side lengths and/or angle measures are known.

solving

angle of elevation

the angle formed by a horizontal line and a line of sight to a point above the horizontal

angle of depression

the angle formed by a horizontal line and a line of sight to a point below the horizontal

trigonometry

the branch of mathematics that deals with the relationships between the sides and the angles of triangles

reference angle

the positive acute angle formed by the terminal side of an angle in standard position and the x-axis

geometric mean

the positive nth root of the product of n factors

trigonometric ratios

the ratios of the lengths of the two sides of a right triangle

used to compare values that are proportional

trig mean

The branch of mathematics that deals with the relationships between the sides and the angles of triangles is called ___.

trigonometry