Introduction to Trigonometry Right Triangle Trigonometry

Introduction What special theorem do you already know that applies to a right triangle? Pythagorean Theorem: a2 + b2 = c2 c a b

Introduction Trigonometry is a branch of mathematics that uses right triangles to help you solve problems. Trig is useful to surveyors, engineers, navigators, and machinists (and others too.)

Finding Trigonometric Ratios A trigonometric ratio is a ratio of the lengths of two sides of a right triangle. The word trigonometry is derived from the ancient Greek language and means measurement of triangles. The three basic trigonometric ratios are sine, cosine, and tangent, which are abbreviated as sin, cos, and tan, respectively.

Topic 1 Before we can understand the trigonometric ratios we need to know how to label Right Triangles.

Labeling Right Triangles The most important skill you need right now is the ability to correctly label the sides of a right triangle. The names of the sides are: the hypotenuse the opposite side the adjacent side

Labeling Right Triangles The hypotenuse is easy to locate because it is always found across from the right angle. Since this side is across from the right angle, this must be the hypotenuse. Here is the right angle...

Labeling Right Triangles Before you label the other two sides you must have a reference angle selected. It can be either of the two acute angles. In the triangle below, let’s pick angle B as the reference angle. A B C This will be our reference angle...

Labeling Right Triangles Remember, angle B is our reference angle. The hypotenuse is side BC because it is across from the right angle. A B (ref. angle) C hypotenuse

Labeling Right Triangles Side AC is across from our reference angle B. So it is labeled: opposite. A B (ref. angle) C hypotenuse opposite

Labeling Right Triangles Adjacent means beside or next to The only side unnamed is side AB. This must be the adjacent side. A B (ref. angle) C adjacent hypotenuse opposite

Labeling Right Triangles Let’s put it all together. Given that angle B is the reference angle, here is how you must label the triangle: A B (ref. angle) C hypotenuse adjacent opposite

Labeling Right Triangles Given the same triangle, how would the sides be labeled if angle C were the reference angle? Will there be any difference?

Labeling Right Triangles Angle C is now the reference angle. Side BC is still the hypotenuse since it is across from the right angle. B hypotenuse C (ref. angle) A

Labeling Right Triangles However, side AB is now the side opposite since it is across from angle C. B opposite hypotenuse C (ref. angle) A

Labeling Right Triangles That leaves side AC to be labeled as the adjacent side. B hypotenuse opposite C (ref. angle) A adjacent

Labeling Right Triangles Let’s put it all together. Given that angle C is the reference angle, here is how you must label the triangle: A B C (ref. angle) hypotenuse opposite adjacent

Labeling Practice Given that angle X is the reference angle, label all three sides of triangle WXY. Do this on your own. Click to see the answers when you are ready. W X Y

Labeling Practice How did you do? Click to try another one... W X Y adjacent opposite hypotenuse

Labeling Practice Given that angle R is the reference angle, label the triangle’s sides. Click to see the correct answers. R S T

Labeling Practice The answers are shown below: R T S hypotenuse adjacent opposite

Finding Trigonometric Ratios A trigonometric ratio is a ratio of the lengths of two sides of a right triangle. TRIGONOMETRIC RATIOS Let be a right triangle. The sine, the cosine, and the tangent of the acute angle A are defined as follows. ABC side opposite A hypotenuse = o h B sin A = hypotenuse side opposite A h o side adjacent A hypotenuse = a h cos A = A C a side opposite A side adjacent to A = o a tan A = side adjacent to A The value of the trigonometric ratio depends only on the measure of the acute angle, not on the particular right triangle that is used to compute the value.

Finding Trigonometric Ratios Compare the sine, the cosine, and the tangent ratios for A in each triangle below. A B C 8.5 4 7.5 SOLUTION These triangles were created so that  A has the same measurement in both triangles. A B C 17 8 15 Large triangle Small triangle sin A = opposite hypotenuse  0.4706 8 17  0.4706 4 8.5 cos A = adjacent hypotenuse  0.8824 15 17  0.8824 7.5 8.5 tan A = opposite adjacent  0.5333 4 7.5  0.5333 8 15 Trigonometric ratios are frequently expressed as decimal approximations. The ratios of the sides for a certain angle size stays constant. We can use this to help us find missing sides and missing angles.

How do I remember these?

Finding Trigonometric Ratios Find the sine, the cosine, and the tangent of the indicated angle. R T S 5 13 12 S SOLUTION The length of the hypotenuse is 13. For S, the length of the opposite side is 5, and the length of the adjacent side is 12. opp. hyp. = = 5 13 sin S  0.3846 R T S 5 12 13 adj. hyp. = = 12 13 hyp. cos S  0.9231 opp. opp. adj. = = 5 12 adj. tan S  0.4167

Finding Trigonometric Ratios Find the sine, the cosine, and the tangent of the indicated angle. R T S 5 13 12 R SOLUTION The length of the hypotenuse is 13. For R, the length of the opposite side is 12, and the length of the adjacent side is 5. opp. hyp. = = 12 13 sin R  0.9231 R T S 5 12 13 adj. hyp. = = 5 13 hyp. cos R  0.3846 adj. opp. adj. = = 12 5 opp. tan R = 2.4

Trigonometric Ratios for 45º Find the sine, the cosine, and the tangent of 45º. SOLUTION Because all such triangles are similar, you can make calculations simple by choosing 1 as the length of each leg. From the 45º-45º-90º Triangle Theorem, it follows that the length of the hypotenuse is 2 . opp. hyp. = = 1 2 = 2 sin 45º  0.7071 45º 2 hyp. 1 adj. hyp. = = 1 2 = 2 cos 45º  0.7071 1 opp. adj. = = 1 tan 45º = 1

Trigonometric Ratios for 30º Find the sine, the cosine, and the tangent of 30º. SOLUTION To make the calculations simple, you can choose 1 as the length of the shorter leg. From the 30º-60º-90º Triangle Theorem, it follows that the length of the longer leg is 3 and the length of the hypotenuse is 2. opp. hyp. = = 1 2 sin 30º = 0.5 30º 2 adj. hyp. = 3 2 = cos 30º  0.8660 1 opp. adj. = = 1 3 3 = tan 30º  0.5774 3

Sample keystroke sequences Sample calculator display Rounded Using a Calculator You can use a calculator to approximate the sine, the cosine, and the tangent of 74º. Make sure your calculator is in degree mode. The table shows some sample keystroke sequences accepted by most calculators. Sample keystroke sequences Sample calculator display Rounded approximation 74 or sin Enter 0.961261695 0.9613 74 or cos Enter 0.275637355 0.2756 74 or tan Enter 3.487414444 3.4874

Finding Trigonometric Ratios The sine or cosine of an acute angle is always less than 1. The reason is that these trigonometric ratios involve the ratio of a leg of a right triangle to the hypotenuse. The length of a leg of a right triangle is always less than the length of its hypotenuse, so the ratio of these lengths is always less than one. Because the tangent of an acute angle involves the ratio of one leg to another leg, the tangent of an angle can be less than 1, equal to 1, or greater than 1.

Finding Trigonometric Ratios The sine or cosine of an acute angle is always less than 1. The reason is that these trigonometric ratios involve the ratio of a leg of a right triangle to the hypotenuse. The length of a leg of a right triangle is always less than the length of its hypotenuse, so the ratio of these lengths is always less than one. Because the tangent of an acute angle involves the ratio of one leg to another leg, the tangent of an angle can be less than 1, equal to 1, or greater than 1. TRIGONOMETRIC IDENTITIES A trigonometric identity is an equation involving trigonometric ratios that is true for all acute angles. The following are two examples of identities: (sin A) 2 + (cos A) 2 = 1 A B C c a b tan A = sin A cos A

Using Trigonometric Ratios in Real Life Suppose you stand and look up at a point in the distance, such as the top of the tree. The angle that your line of sight makes with a line drawn horizontally is called the angle of elevation.

The tree is about 75 feet tall. Indirect Measurement FORESTRY You are measuring the height of a Sitka spruce tree in Alaska. You stand 45 feet from the base of a tree. You measure the angle of elevation from a point on the ground to the top of the tree to be 59°. To estimate the height of the tree, you can write a trigonometric ratio that involves the height h and the known length of 45 feet. tan 59° = opposite adjacent Write ratio. tan 59° = opposite adjacent h 45 Substitute. 45 tan 59° = h Multiply each side by 45. 45(1.6643)  h Use a calculator or table to find tan 59°. 74.9  h Simplify. The tree is about 75 feet tall.

A person travels 152 feet on the escalator stairs. Estimating a Distance ESCALATORS The escalator at the Wilshire/Vermont Metro Rail Station in Los Angeles rises 76 feet at a 30° angle. To find the distance d a person travels on the escalator stairs, you can write a trigonometric ratio that involves the hypotenuse and the known leg length of 76 feet. sin 30° = opposite hypotenuse Write ratio for sine of 30°. sin 30° = opposite hypotenuse 76 d Substitute. 30° 76 ft d d sin 30° = 76 Multiply each side by d. d = 76 sin 30° Divide each side by sin 30°. d = 76 0.5 Substitute 0.5 for sin 30°. d = 152 Simplify. A person travels 152 feet on the escalator stairs.

Homework Pg. 469 #3,4,13 Pg. 477 # 3,4,6,7,8