2.  Old stuff will be used in this section › Triangle Sum Theorem  The sum of the measures of the angles in a triangle is 180° › Pythagorean Theorem  In a right triangle with legs a and b and hypotenuse c, a 2 + b 2 = c 2

3.  Finding a side of a Triangle › Find side x in the right triangle below  In this figure, we’re given: An angle (65°) The H ypotenuse (8) A side A djacent to 65° (x) The sides we’re using are A and H using SOH- CAH -TOA means we use the cosine function 65° 8 x

4.  Finding an Angle of a Triangle › Find the measure of the angle θ in the triangle below In this triangle, we’re given all three side lengths, so we can use any of the trigonometric ratios to solve. › SOH  sin θ = 3/5 → sin -1 (3/5) = 36.8699° › CAH  cos θ = 4/5 → cos -1 (4/5) = 36.8699° › TOA  tan θ = 3/4 → tan -1 (3/4) = 36.8699° › All ratios give us the same answer: 36.8699° θ 4 5 3

5.  Solving a Right Triangle › Solve the right triangle below  The Triangle Sum Theorem helps find θ 75° + θ + 90° = 180° θ = 15° We can use the hypotenuse (17) and the 75° angle to find sides a and b 75° 17 b a θ

6.  Solving a Right Triangle › Solve the right triangle below  The Pythagorean Theorem helps find a a 2 + 6 2 = 12 2 a 2 = 108 a = We can find β by using the cosine function cos β = 6/12 cos β = 1/2 cos -1 (1/2) = β 60° = β We can either find θ by using the sin function or by using The Triangle Sum Theorem θ = 30° β 12 6 a θ

7.  Assignment › Page 429  Problems 1 – 35, odd problems  Questions where you’re told to not use a calculator can be solved using the chart you copied yesterday.

9.  Applications › A straight road leads from an ocean beach at a constant upward angle of 3°. How high above sea level is the road at a point 1 mile from the beach?  Answer  If one is not drawn, DRAW A DIAGRAM.   Looking for the side on the right of the triangle, which is the side opposite of 3°  sin 3° = x/5280 5280 sin 3° = x 276.33 ft = x 3°3° h

10.  Applications › According to the safety sticker on a 20-foot ladder, the distance from the bottom of the ladder to the base of the wall on which it leans should be one-fourth of the length of the ladder: 5 feet.  How high up the wall will the ladder reach  If the ladder is in this position, what angle does it make with the ground?  Draw a diagram ground 5 ft wall h ft ladder 20 ft θ

11.  Applications › The wall height can be found using the Pythagorean Theorem  5 2 + h 2 = 20 2  h 2 = 400 – 25  h 2 = 375  h = (375) ½ ≈ 19.36 ft › We’re given the side adjacent to θ and the hypotenuse, meaning we need to use cosine  cos θ = 5/20  θ = cos -1 (5/20) ≈ 75.5° ground 5 ft wall h ft ladder 20 ft θ

12.  Angles of Elevation and Depression › Both create right angles from an endpoint. Angles of elevation look up; angles of depression look down. Angle of elevation Angle of depression

13.  Elevation/Depression › A flagpole casts a 60-foot shadow when the angle of elevation of the sun is 35°. Find the height of the flagpole.  Draw a diagram  You’re given a 35° angle  You’re given the side adjacent  You’re looking for the side opposite  You’re using tangent  tan 35° = x / 60  60 tan 35° = x  42.012 ≈ x shadow 60 ft flagpole h ft 35°

14.  Elevation/Depression (#3: both) › A person on the edge of a canal observes a lamp post on the other side with an angle of elevation of 12° to the top of the lamp post and an angle of depression of 7° to the bottom of the lamp post from eye level. The person’s eye level is 152 cm.  Find the width of the canal.  Find the height of the lamp post.  Draw a diagram 12° 7° 152 cm canal top half of lamp post

15.  Elevation/Depression (#3: both) › The canal is adjacent to the 7° angle › You’re given 152 cm, which is opposite 7°  Use tangent  tan 7° = 152 / x  x = 152 / tan 7°  x = 1237.94 cm › Use the canal measurement to find the top half of the lamp post… again using tangent.  tan 12° = y / 1237.94  1237.94 tan 12° = y  263.13 cm = y › So the height of the lamp post is 263.13 + 152 = 415.13 cm 12° 7° 152 cm canal (x) top half of lamp post (y)

16.  Assignment › Page 431  Problems 37 – 49, odd problems › Quiz tomorrow 1)DMS/decimal conversion 2)Finding 6 trig ratios 3)Solving right triangles 4)A word problem or two