Right Triangle Trigonometry

1. Right Triangle Trigonometry Section 6.5

2. 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

3. 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

4. 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

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

6. 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.

7. 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

8. 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

9. 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°

10. 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

11. 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°

12. Example • Find the missing side lengths and angles. 60° A = 180°-90°-60°=30° sin(60°)=y/10 10 x thus y=10sin(60°) A y

13. 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.

14. 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.

15. 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)

16. 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