Right Triangle Trigonometry Lesson 7-4 Right Triangle Trigonometry Modified by Lisa Palen (Reference Angle instead of Angle of Perspective) With slides by Mr. Jerrell Walker, Lincoln, Nebraska and Emily Freeman, Powder Springs, Georgia.

Anatomy of a Right Triangle hypotenuse hypotenuse opposite opposite adjacent adjacent reference angle

If the reference angle is then then Hyp Adj Hyp Opp Opp Adj Right Triangle Trigonometry

The Trigonometric Functions we will be looking at SINE COSINE TANGENT

The Trigonometric Functions SINE COSINE TANGENT

SINE Prounounced “sign”

Prounounced “co-sign” COSINE Prounounced “co-sign”

Prounounced “tan-gent”

Greek Letter q Prounounced “theta” Represents an unknown angle Sometimes called the reference angle or angle of perspective

Definitions of Trig Ratios hypotenuse hypotenuse opposite opposite adjacent adjacent

We need a way to remember all of these ratios…

Sin SOHCAHTOA Opp Hyp Cos Adj Hyp Tan Opp Adj Old Hippie

Some Old Hippie CAught Another Hippy Tripping On Acid Old Hippie

Finding sin, cos, and tan

SOHCAHTOA 10 8 6

10.8 9 A 6 Find the sine, the cosine, and the tangent of angle A. Give a fraction and decimal answer (round to 4 places). 10.8 9 A 6

? 5 4 3 Pythagorean Theorem: (3)² + (4)² = c² 5 = c Find the values of the three trigonometric functions of . ? Pythagorean Theorem: 5 4 (3)² + (4)² = c² 5 = c 3

24.5 8.2 23.1 Find the sine, the cosine, and the tangent of angle A Give a fraction and decimal answer (round to 4 decimal places). B 24.5 8.2 A 23.1

Finding a side

Example: Find the value of x. Step 1: Identify the “reference angle”. Step 2: Label the sides (Hyp / Opp / Adj). Step 3: Select a trigonometry ratio (sin/ cos / tan). Sin = Step 4: Substitute the values into the equation. Sin 25 = Step 5: Solve the equation. Hyp opp reference angle Adj x = 12 sin 25 x = 12 (.4226) x = 5.07 cm = Right Triangle Trigonometry

Solving Trigonometric Equations There are only three possibilities for the placement of the variable ‘x”. Sin = sin = sin = sin 25 = sin x = sin 25 = sin 25 = x = sin (12/25) sin 25 = x = x = 28.7 x = (12) (sin 25) x = 5.04 cm x = 28.4 cm Note you are looking for an angle here!

Angle of Elevation and Depression Example #1

Angle of Elevation and Depression Suppose the angle of depression from a lighthouse to a sailboat is 5.7o. If the lighthouse is 150 ft tall, how far away is the sailboat? 5.7o 150 ft. 5.7o x Construct a triangle and label the known parts. Use a variable for the unknown value.

Angle of Elevation and Depression Suppose the angle of depression from a lighthouse to a sailboat is 5.7o. If the lighthouse is 150 ft tall, how far away is the sailboat? 5.7o 150 ft. 5.7o x Set up an equation and solve.

Angle of Elevation and Depression 150 ft. 5.7o Remember to use degree mode! x x is approximately 1,503 ft.

Angle of Elevation and Depression Example #2

Angle of Elevation and Depression A spire sits on top of the top floor of a building. From a point 500 ft. from the base of a building, the angle of elevation to the top floor of the building is 35o. The angle of elevation to the top of the spire is 38o. How tall is the spire? Construct the required triangles and label. 38o 35o 500 ft.

Angle of Elevation and Depression Write an equation and solve. Total height (t) = building height (b) + spire height (s) Solve for the spire height. s t Total Height b 38o 35o 500 ft.

Angle of Elevation and Depression Write an equation and solve. Building Height s t b 38o 35o 500 ft.

Angle of Elevation and Depression Write an equation and solve. Total height (t) = building height (b) + spire height (s) s t b The height of the spire is approximately 41 feet. 38o 35o 500 ft.

Angle of Elevation and Depression Example #3

Angle of Elevation and Depression A hiker measures the angle of elevation to a mountain peak in the distance at 28o. Moving 1,500 ft closer on a level surface, the angle of elevation is measured to be 29o. How much higher is the mountain peak than the hiker? Construct a diagram and label. 1st measurement 28o. 2nd measurement 1,500 ft closer is 29o.

Angle of Elevation and Depression Adding labels to the diagram, we need to find h. h ft 29o 28o 1500 ft x ft Write an equation for each triangle. Remember, we can only solve right triangles. The base of the triangle with an angle of 28o is 1500 + x.

Angle of Elevation and Depression Now we have two equations with two variables. Solve by substitution. Solve each equation for h. Substitute.

Angle of Elevation and Depression Solve for x. Distribute. Get the x’s on one side and factor out the x. Divide. x = 35,291 ft.

Angle of Elevation and Depression x = 35,291 ft. However, we were to find the height of the mountain. Use one of the equations solved for “h” to solve for the height. The height of the mountain above the hiker is 19,562 ft.

Ex. 4 A surveyor is standing 50 feet from the base of a large tree. The surveyor measures the angle of elevation to the top of the tree as 71.5°. How tall is the tree? tan 71.5° ? tan 71.5° 71.5° y = 50 (tan 71.5°) 50 y = 50 (2.98868)

Ex. 5 A person is 200 yards from a river. Rather than walk directly to the river, the person walks along a straight path to the river’s edge at a 60° angle. How far must the person walk to reach the river’s edge? cos 60° x (cos 60°) = 200 200 60° x x X = 400 yards