Right Triangle Trig Section 4.3

Right Triangle Trig In the previous section, we worked with: Angles in both radians & degrees Coterminal angles Complementary and supplementary angles Linear and angular speed Now our focus is going to shift to triangles

Right Triangle Trig In this section, we are going to be using only right triangles Opp. Hyp. Ѳ Adj.

Right Triangle Trig In this section, we are going to be using only right triangles Ѳ Adj. Hyp. Opp.

Right Triangle Trig Using these three sides of the right triangle, we can form six ratios that define the six trigonometric functions. 1. Sine θ= Opposite Hypotenuse 4. Cosecant θ= Hyp. Opp. 2. Cosine θ= Adjacent Hypotenuse 5. Secant θ= Hyp. Adj. 3. Tangent θ= Opposite Adjacent 6. Cotangent θ= Adj. Opp.

Right Triangle Trig S o h C a h T o a Sine = Opposite / Hypotenuse Cosine = Adjacent / Hypotenuse Tangent = Opposite / Adjacent S o h C a h T o a

Right Triangle Trig Find the value of the six trig functions of the following triangle. 3 Ѳ 4

Right Triangle Trig Find the value of the six trig functions of the following triangle. Ѳ 17 15

Right Triangle Trig Find the value of the six trig functions of the following triangle. 5 Ѳ 8

Right Triangle Trig You can also use a trig value to construct a right triangle and find the values of the remaining trig functions. E.g. Sin 𝜃= 12 13

Right Triangle Trig Sketch a right triangle and find the values of the remaining trig functions using the given information. Cos 𝜃= 7 25 Tan 𝜃=3 Csc 𝜃= 3 4

Right Triangle Trig Cos 𝜃= 7 25

Right Triangle Trig Tan 𝜃=3

Right Triangle Trig Csc 𝜃= 3 4

Right Triangle Trig Section 4.3

Right Triangle Trig Yesterday: Today Defined all six trig functions Found values of all six trig functions from right triangles Constructed right triangles from a specific trig value and found the remaining values Today Special right triangles Using a calculator

Right Triangle Trig Sketch a right triangle and find the values of the remaining trig functions using the given information. Cot 𝜃=1

Right Triangle Trig Find the sine, cosine, and tangent of 45˚

Right Triangle Trig Using the equilateral triangle below, find the sine, cosine, and tangent of both 30˚ and 60˚. 2

Right Triangle Trig These triangles are our two special triangles. We will use them throughout the year. The sooner you remember them, the easier your life will be.

Right Triangle Trig 1 2 2 Csc 30˚ = Sin 30˚ = 2 3 3 3 2 Sec 30˚ = From your triangles: 1 2 2 Sin 30˚ = Csc 30˚ = 3 2 2 3 3 Cos 30˚ = Sec 30˚ = 3 3 3 Tan 30˚ = Cot 30˚ =

Right Triangle Trig 3 2 2 3 3 Sin 60˚ = Csc 60˚ = 1 2 Cos 60˚ = From your triangles: 3 2 2 3 3 Sin 60˚ = Csc 60˚ = 1 2 Cos 60˚ = Sec 60˚ = 2 3 3 3 Tan 60˚ = Cot 60˚ =

Right Triangle Trig 2 2 2 Sin 45˚ = Csc 45˚ = 2 2 2 Cos 45˚ = From your triangles: 2 2 2 Sin 45˚ = Csc 45˚ = 2 2 2 Cos 45˚ = Sec 45˚ = 1 1 Tan 45˚ = Cot 45˚ =

Right Triangle Trig The trig values of the angles 30˚, 45˚, and 60˚ are values that we will be using from now until May. You will be expected to know these values from memory. There will be quizzes that are non-calculator where these values will be needed.

Right Triangle Trig Using a calculator On your calculators, you should see the three main trig functions. Using these buttons, find the Sine 10˚

Right Triangle Trig Your calculator can also evaluate trig functions in radians. To do this, you must switch the mode from degrees to radians. Find the Cos 3𝜋 2

Right Triangle Trig Using your calculator, evaluate the following: Tan 67˚ Sin 3.4 Sec 35˚ Cot 4𝜋 5

Right Triangle Trig In addition to radians and degrees, there is one more type of unit we will be using throughout the year. Minutes and Seconds Most commonly used in longitude and latitude An angle in minutes and seconds would look like: 56˚ 8́ 10˝

Right Triangle Trig To convert minutes and seconds to degrees: 56˚ 8́ 10˝ Find the sum of the whole angle, the minutes divided by 60, and the seconds divided by 3,600 This will give you your angle in degrees

Right Triangle Trig Evaluate the following trig functions: Sin 73˚ 56́ Tan 44˚ 28́ 16˝ Sec 4˚ 50́ 15˝

Right Triangle Trig Section 4.3

Right Triangle Trig Find the remaining five trig functions if Tan Ѳ = 5 12 Find the exact value of the Cos 60˚ and Csc 𝜋 3 Evaluate the Sec 37˚ to three decimal places

Right Triangle Trig So far: Today Defined the six trig functions Created triangles from given trig values to find the remaining trig values Used the special right triangles 30-60-90 45-45-90 Used a calculator to evaluate trig functions of other angles Converted between degrees and minutes/seconds Today Find angles when given trig values Fundamental Identities

Right Triangle Trig So far, we have been using our trig functions to create ratios. We can also use trig functions to solve for entire triangles when given certain information. Must be given 1 side and one other part of the triangle.

Right Triangle Trig Solve for the remaining parts of the triangle. 8 35˚

Right Triangle Trig Solve for the remaining parts of the triangle. 20˚ 12

Right Triangle Trig Solve for the remaining parts of the triangle. 5˚ 18

Right Triangle Trig So far, the information we have been given has been 1 side and 1 angle. When we are given 2 sides, you must use your calculator to evaluate the angle. This involves the inverse trig buttons on your calc. Sin −1 Cos −1 Tan −1

Right Triangle Trig Solve for the remaining parts of the triangle. 10 7

Right Triangle Trig Solve for the remaining parts of the triangle. 13 9

Right Triangle Trig Use the given information to solve for the remaining parts of each triangle. 22˚ 7 11 4

Right Triangle Trig 7 4

Right Triangle Trig 22˚ 11

Right Triangle Trig Fundamental Trig Identities These are identities that we will use throughout the year It will be very beneficial to memorize them now as opposed to struggling to remember them later We will go over 11 now, there will be over 30 throughout the course of the year These are on page 283 of your book

Right Triangle Trig = 1 Csc θ = 1 Csc θ = 1 Csc θ Sin θ Cos θ Tan θ Reciprocal Identities = 1 Csc θ = 1 Csc θ = 1 Csc θ Sin θ Cos θ Tan θ = 1 Sin θ = 1 Sin θ = 1 Sin θ Csc θ Sec θ Cot θ

Right Triangle Trig = Sin θ Cos θ = Cos θ Sin θ Tan θ Cot θ Quotient Identities = Sin θ Cos θ = Cos θ Sin θ Tan θ Cot θ

Right Triangle Trig Sin 2 θ+ Cos 2 θ=1 1+Tan 2 θ= Sec 2 θ Pythagorean Identities Sin 2 θ+ Cos 2 θ=1 1+Tan 2 θ= Sec 2 θ 1+Cot 2 θ= Csc 2 θ

Right Triangle Trig We use the fundamental identities for 2 main purposes: To evaluate trig functions when given certain information Transformations (proofs)

Right Triangle Trig Let Ѳ be an acute angle such that Sin Ѳ = 0.6. Use the fundamental identities to find the Cos Ѳ and Tan Ѳ.

Right Triangle Trig If Tan Ѳ = 5, find the remaining five trig functions of Ѳ.

Right Triangle Trig If Csc Ѳ = 13 2 , find the remaining 5 trig functions of Ѳ.

Right Triangle Trig Transformations Similar to proofs from geometry Do not need to list reasons, just show steps Can only work with 1 side of the equation For this section, we will only be working with the left side of the equation.

Right Triangle Trig Use the fundamental identities to transform the left side of the equation into the right side. Cos Ѳ Sec Ѳ = 1

Right Triangle Trig Use the fundamental identities to transform the left side of the equation into the right side. Cot Ѳ Sin Ѳ = Cos Ѳ

Right Triangle Trig Use the fundamental identities to transform the left side of the equation into the right side. (1 + Sin Ѳ) (1 – Sin Ѳ) = Cos 2 Ѳ

Right Triangle Trig Tan θ+Cot θ Tan θ = Csc 2 θ Use the fundamental identities to transform the left side of the equation into the right side. Tan θ+Cot θ Tan θ = Csc 2 θ

Right Triangle Trig When dealing with word problems: Start by drawing a picture (preferably relating to the problem) Label all sides of the triangle Determine what information you are looking for Set up a trigonometric ratio and solve

Right Triangle Trig Suppose you are standing parallel to the Concord River. You turn and walk at a 60 degree angle until you reach the river, which is a total distance of 130 ft. How much farther down the river are you than when you started?

Right Triangle Trig You are playing paintball with a friend when you notice him sitting in a tree. The angle of elevation to your friend is 23°. If the distance between yourself and the tree is 60 ft, how high up in the tree is your friend?

Right Triangle Trig Suppose you are putting in a basketball hoop and need to know where to put the “foul line”. You know that the height of the hoop is 10 ft and the angle of elevation from the “foul line” is 33.7°. Use this information to determine where the foul line should go.

Right Triangle Trig You are a surveyor standing 115 ft from the Washington Monument in Washington D.C. You measure the angle of elevation to the top of the monument to be 78.3°. How tall is the Washington Monument?

Right Triangle Trig