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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.
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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.
Right Triangle Trig • 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
Right Triangle Trig • Sketch a right triangle and find the values of the remaining trig functions using the given information. • Cos • Tan • Csc
Right Triangle Trig • Cos
Right Triangle Trig • Tan
Right Triangle Trig • Csc
Right Triangle Trig Section 4.3
Right Triangle Trig • Yesterday: • 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
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 • From your triangles: 2 = = = = = =
Right Triangle Trig • From your triangles: = = 2 = = = =
Right Triangle Trig • From your triangles: = = = = 1 = =
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
Right Triangle Trig • Using your calculator, evaluate the following: • Tan 67˚ • Sin 3.4 • Sec 35˚ • Cot
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 = • Find the exact value of the Cos 60˚ and Csc • Evaluate the Sec 37˚ to three decimal places
Right Triangle Trig • So far: • 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.
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 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 • Reciprocal Identities = = = = = =
Right Triangle Trig • Quotient Identities = =
Right Triangle Trig • Pythagorean Identities
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.
