Right Triangle Trigonometry Section 6.5

Pythagorean Theorem Recall that a right triangle has a 90° angle as one of its angles. The side that is opposite the 90° angle is called the hypotenuse. The theorem due to Pythagoras says that the square of the hypotenuse is equal to the sum of the squares of the legs. c2 = a2 + b2 a c b

Similar Triangles Triangles are similar if two conditions are met: The corresponding angle measures are equal. Corresponding sides must be proportional. (That is, their ratios must be equal.) The triangles below are similar. They have the same shape, but their size is different. A D c b f e E d F B a C

Corresponding angles and sides As you can see from the previous page we can see that angle A is equal to angle D, angle B equals angle E, and angle C equals angle F. The lengths of the sides are different but there is a correspondence. Side a is in correspondence with side d. Side b corresponds to side e. Side c corresponds to side f. What we do have is a set of proportions. a/d = b/e = c/f

Example Find the missing side lengths for the similar triangles. 3.2 3.8 y 54.4 x 42.5

ANSWER Notice that the 54.4 length side corresponds to the 3.2 length side. This will form are complete ratio. To find x, we notice side x corresponds to the side of length 3.8. Thus we have 3.2/54.4 = 3.8/x. Solve for x. Thus x = (54.4)(3.8)/3.2 = 64.6 Same thing for y we see that 3.2/54.4 = y/42.5. Solving for y gives y = (42.5)(3.2)/54.4 = 2.5.

Introduction to Trigonometry In this section we define the three basic trigonometric ratios, sine, cosine and tangent. opp is the side opposite angle A adj is the side adjacent to angle A hyp is the hypotenuse of the right triangle hyp opp adj A

Definitions Sine is abbreviated sin, cosine is abbreviated cos and tangent is abbreviated tan. The sin(A) = opp/hyp The cos(A) = adj/hyp The tan(A) = opp/adj Just remember sohcahtoa! Sin Opp Hyp Cos Adj Hyp Tan Opp Adj

Special triangles 30 – 60 – 90 degree triangle. Consider an equilateral triangle with side lengths 2. Recall the measure of each angle is 60°. Chopping the triangle in half gives the 30 – 60 – 90 degree traingle. 30° 2 2 √3 2 2 1 60°

30° – 60° – 90° Now we can define the sine cosine and tangent of 30° and 60°. sin(60°)=√3 / 2; cos(60°) = ½; tan(60°) = √3 sin(30°) = ½ ; cos(30°) = √3 / 2; tan(30°) = 1/√3

45° – 45° – 90° Consider a right triangle in which the lengths of each leg are 1. This implies the hypotenuse is √2. 45° sin(45°) = 1/√2 √2 cos(45°) = 1/√2 1 tan(45°) = 1 1 45°

Example Find the missing side lengths and angles. sin(60°)=y/10 10 x thus y=10sin(60°) A y

Inverse Trig Functions What if you know all the sides of a right triangle but you don’t know the other 2 angle measures. How could you find these angle measures? What you need is the inverse trigonometric functions. Think of the angle measure as a present. When you take the sine, cosine, or tangent of that angle, it is similar to wrapping your present. The inverse trig functions give you the ability to unwrap your present and to find the value of the angle in question.

Notation A=sin-1(z) is read as the inverse sine of A. Never ever think of the -1 as an exponent. It may look like an exponent and thus you might think it is 1/sin(z), this is not true. (We refer to 1/sin(z) as the cosecant of z) A=cos-1(z) is read as the inverse cosine of A. A=tan-1(z) is read as the inverse tangent of A.

Inverse Trig definitions Referring to the right triangle from the introduction slide. The inverse trig functions are defined as follows: A=sin-1(opp/hyp) A=cos-1(adj/hyp) A=tan-1(opp/adj)

Example using inverse trig functions Find the angles A and B given the following right triangle. Find angle A. Use an inverse trig function to find A. For instance A=sin-1(6/10)=36.9°. Then B = 180° - 90° - 36.9° = 53.1°. B 6 10 8 A