Right Triangle Trigonometry

Right Triangle Trigonometry Section 4.3

Objectives • Calculate any trigonometric function for an angle in a right triangle given two sides of the triangle. • Calculate the length of the sides of a right triangle given the measure of an angle of a triangle. • Solve word problems requiring right triangles and trigonometric functions.

Vocabulary • angle of elevation • angle of depression the angle that an observer would raise his or her line of sight above a horizontal line in order to see an object. angle of elevation If an observer were UP ABOVE and needed to look down, the angle of depression would be the angle that the person would need to lower his or her line of sight. angle of depression

hypotenuse opposite (for angle x) x adjacent (for angle x) Trigonometric Functions Each of the sides of a right triangle can be labeled relative to one of the non-right angles of the triangle. The side of the triangle opposite the right angle is always named the hypotenuse. If we label one of the non-right angles x, then we can name the legs of the triangle. The leg that makes up part of the angle is called the adjacent side. The leg that is not one of the sides of the angle is called the oppositeside.

Trigonometric Functions The names of the sides will change if we change the angle. If we use the other non-right angle in the triangle (angle y), then the sides that are opposite and adjacent change. The side that is the hypotenuse never changes. It is always opposite the right angle. hypotenuse y adjacent (for angle y) opposite (for angle y)

hypotenuse opposite (for angle x) x adjacent (for angle x) Trigonometric Functions Once we have named our sides, we can define each of the trigonometric functions as ratios of the sides of the triangle.

hypotenuse opposite (for angle x) x adjacent (for angle x) Trigonometric Functions • sin(x) • cos(x) • tan(x)

hypotenuse opposite (for angle x) x adjacent (for angle x) Trigonometric Functions • csc(x) • sec(x) • cot(x)

For the triangle y • Find sin(x) 3 c To find the sin(x), we will need to length of the side opposite the angle x and the length of the hypotenuse. We have the length of the opposite side, 3. To find the length of the hypotenuse, we can use the Pythagorean Theorem. x 5 continued on next slide

For the triangle y Now that we have the length of the hypotenuse, we can answer all of the questions. • Find sin(x) 3 x 5 continued on next slide

For the triangle y Notice that for this question the angle has changed to angle y. This will change what is opposite and what is adjacent. • Find cos(y) 3 x 5 One thing to note here is that angle y is equal to 90 - x. It is also the case that x is equal to 90 – y. the In general when this happens, we have the following co-function identities: and continued on next slide

For the triangle y Now continuing with the tangent and cotangent functions. • Find tan(x) 3 x 5 continued on next slide

For the triangle y Now continuing with the tangent and cotangent functions. • Find cot(y) 3 x 5 One thing to note here is that angle y is equal to 90 - x. It is also the case that x is equal to 90 – y. the In general when this happens, we have the following co-function identities: and continued on next slide

For the triangle y Now continuing with the secant and cosecant functions. • Find sec(x) 3 x 5 continued on next slide

For the triangle y Now continuing with the secant and cosecant functions. • Find sec(x) 3 x 5 One thing to note here is that angle y is equal to 90 - x. It is also the case that x is equal to 90 – y. the In general when this happens, we have the following co-function identities: and

Identities • Cofunction Identities • sin(x) = cos(90-x) • cos(x) = sin(90-x) • tan(x) = cot(90-x) • cot(x) = tan(90-x) • sec(x) = csc(90-x) • csc(x) = sec(90-x)

C β α B A For the triangle below, if BC = 7 and the angle β = 60, find all the missing angles and sides. Let’s start by putting the information that we know into the triangle picture. 60° 7 continued on next slide

C β α B A For the triangle below, if BC = 7 and the angle β = 60, find all the missing angles and sides. Our next step will be to do the easiest work. I think that the easiest thing to find next is the measure of the angle α. Since the measure of the angles of a triangle add up to 180, we can calculate α as α=180°-60°-90°=30° We can now put this into the triangle. 60° 7 30° continued on next slide

C β α B A For the triangle below, if BC = 7 and the angle β = 60, find all the missing angles and sides. Now we need to find lengths of sides of the triangle. It does not matter which side we decide to find first. Let’s find side AB. To do this we need to use one of the non-right angles. Either angle will work. I am going to choose the 60° angle since that was the one that was given. Side AB is opposite the 60 ° angle. The other side that we know is BC which is adjacent to the 60 ° angle. We need the trigonometric function which has both opposite and adjacent in it. This function is the tangent function. 60° 7 30° continued on next slide

C β α B A For the triangle below, if BC = 7 and the angle β = 60, find all the missing angles and sides. We will use the tan(60) to find side AB. 60° 7 30° 12.12435565 continued on next slide

C β α B A For the triangle below, if BC = 7 and the angle β = 60, find all the missing angles and sides. Now we are ready to find side AC. No matter which angle we use, the side AC is the hypotenuse of the triangle. You may be tempted to use the Pythagorean Theorem here, but I do not recommend it. If you have made an error or just rounded the length of the side that was just found, the Pythagorean Theorem will produce an incorrect answer. Since we want to use numbers that were not calculated, but instead given in the problem, we will use the 60° and the side BC = 7. 60° 7 30° 12.12435565 continued on next slide

C β α B A For the triangle below, if BC = 7 and the angle β = 60, find all the missing angles and sides. The side BC = 7 is adjacent to the 60° angle. The side AC is the hypotenuse. We need a trigonometric function which has both the hypotenuse and the adjacent sides. This would be the cosine function. 60° 7 14 30° 12.12435565

A plane if flying at an elevation of 27000 feet. It is within sight of the airport and the pilot finds that the angle of depression to the airport is 25 degrees. Find the distance between the plane and the airport.

A plane if flying at an elevation of 27000 feet. It is within sight of the airport and the pilot finds that the angle of depression to the airport is 25 degrees. Find the distance between the plane and the airport. For this problem we should draw a picture and label what we know. angle of elevation = 25° distance from plane to airport 27000 feet airport distance from a point on the ground directly below the plane to the airport

A plane if flying at an elevation of 27000 feet. It is within sight of the airport and the pilot finds that the angle of depression to the airport is 25 degrees. Find the distance between the plane and the airport. In order to use trigonometric functions, we need a non-right angle inside the triangle. We know that angle and A and the angle of elevation add together to be 90°. Thus we can calculate angle A as A = 90° - 25° = 65°. angle of elevation = 25° A distance from plane to airport 27000 feet airport distance from a point on the ground directly below the plane to the airport continued on next slide

angle of elevation = 25° 65° distance from plane to airport 27000 feet airport distance from a point on the ground directly below the plane to the airport A plane if flying at an elevation of 27000 feet. It is within sight of the airport and the pilot finds that the angle of depression to the airport is 25 degrees. Find the distance between the plane and the airport. Now we can answer the question. For the 65° angle, the side of length 27000 feet is the adjacent side. The side marked as the distance from the plane to the airport is the hypotenuse. A trigonometric function that has both adjacent and hypotenuse is the cosine function. The distance from the plane to the airport is 63887.44275 feet continued on next slide

angle of elevation = 25° 65° distance from plane to airport 27000 feet airport distance from a point on the ground directly below the plane to the airport A plane if flying at an elevation of 27000 feet. It is within sight of the airport and the pilot finds that the angle of depression to the airport is 25 degrees. Find the distance between the plane and the airport. Now we will answer the next question. For the 65° angle, the side of length 27000 feet is the adjacent side. The side marked as the distance from a point on the ground directly below the plane to the airport is the opposite side. A trigonometric function that has both adjacent and opposite is the tangent function. The distance from a point on the ground directly below the plane to the airport is 57901.68685 feet.

A hot-air balloon is floating abovea straight road. To calculate their height above the ground, the balloonists simultaneously measure the angle of depression to two consecutive mileposts on the road on the same side of the balloon. The angles of depression are found to be 23 degrees and 27 degrees. How high (in feet) is the balloon? continued on next slide

23 27 height of the balloon A B 1 mile A hot-air balloon is floating above a straight road. To calculate their height above the ground, the balloonists simultaneously measure the angle of depression to two consecutive mileposts on the road on the same side of the balloon. The angles of depression are found to be 23 degrees and 27 degrees. How high (in feet) is the balloon? Once again we should draw a picture and label what we know. This is not drawn to scale. continued on next slide

23 27 height of the balloon A B 1 mile A hot-air balloon is floating above a straight road. To calculate their height above the ground, the balloonists simultaneously measure the angle of depression to two consecutive mileposts on the road on the same side of the balloon. The angles of depression are found to be 23 degrees and 27 degrees. How high (in feet) is the balloon? The height of the balloon is one of the legs of two different right triangle. One of the triangles is the red one. For this triangle, we do not know the length of the hypotenuse or the length of the other leg. We do however know the length of part of the other leg. continued on next slide

23 27 height of the balloon A B 1 mile A hot-air balloon is floating above a straight road. To calculate their height above the ground, the balloonists simultaneously measure the angle of depression to two consecutive mileposts on the road on the same side of the balloon. The angles of depression are found to be 23 degrees and 27 degrees. How high (in feet) is the balloon? The other of the triangles is the blue one. For this triangle, we do not know the length of the hypotenuse or the length of the other leg. You should note though that the length of the other leg is part of the length of the leg of the red right triangle from the previous screen. That part that the two legs have in common, we can label as x. continued on next slide

23 27 H = height of the balloon A B 1 mile A hot-air balloon is floating above a straight road. To calculate their height above the ground, the balloonists simultaneously measure the angle of depression to two consecutive mileposts on the road on the same side of the balloon. The angles of depression are found to be 23 degrees and 27 degrees. How high (in feet) is the balloon? Now we can work on using a trigonometric function to write an equation for the height of the balloon (we will label this H). We will need to know the measure of the one of the non-right angles. We will use angle A. Angle A and the 23° angle add together to be 90°. This allows us to find A using A = 90 ° - 23° = 67 °. A x continued on next slide

23 27 H = height of the balloon A B 1 mile A hot-air balloon is floating above a straight road. To calculate their height above the ground, the balloonists simultaneously measure the angle of depression to two consecutive mileposts on the road on the same side of the balloon. The angles of depression are found to be 23 degrees and 27 degrees. How high (in feet) is the balloon? The two sides of the triangle that we are interested in are H (adjacent to A) and the side opposite angle A. A trigonometric function that both opposite and adjacent is the tangent function. Thus we can set up the following: A x continued on next slide

23 27 H = height of the balloon A B 1 mile A hot-air balloon is floating above a straight road. To calculate their height above the ground, the balloonists simultaneously measure the angle of depression to two consecutive mileposts on the road on the same side of the balloon. The angles of depression are found to be 23 degrees and 27 degrees. How high (in feet) is the balloon? We can now do the same process with the blue right triangle. We first have to find angle B. Angle B and the 27° angle add up to 90°. Thus we can find B with B = 90 °- 27 °=63 °. Now we can set up the tangent function expression. B x continued on next slide

23 27 H = height of the balloon A 1 mile B A hot-air balloon is floating above a straight road. To calculate their height above the ground, the balloonists simultaneously measure the angle of depression to two consecutive mileposts on the road on the same side of the balloon. The angles of depression are found to be 23 degrees and 27 degrees. How high (in feet) is the balloon? Now we have two equation with two variables. We can solve these using substitution. Since we are interested in finding H, we should solve one equation for x and substitute that into the other equation. Solve this for x. x Now substitute this expression for x into the other equation. continued on next slide

23 27 H = height of the balloon A 1 mile B A hot-air balloon is floating above a straight road. To calculate their height above the ground, the balloonists simultaneously measure the angle of depression to two consecutive mileposts on the road on the same side of the balloon. The angles of depression are found to be 23 degrees and 27 degrees. How high (in feet) is the balloon? Finally we solve for H. x Note that this answer is in the same units as those that your single given distance is in. That distance is miles. We are asked to find the height in feet. There are 5280 feet in 1 mile. Thus the answer is 2.542964269*5280=13426.85134 feet.

The angle of elevation to the top of the Empire State Building in New York is found to be 11 degrees from the ground at a distance of 1 mile from the base of the building. Using this information, find the height of the Empire State Building. continued on next slide

The angle of elevation to the top of the Empire State Building in New York is found to be 11 degrees from the ground at a distance of 1 mile from the base of the building. Using this information, find the height of the Empire State Building. Once again we should draw a picture and label what we know. This is not drawn to scale. Based on this picture we need to find the length of the side opposite the angle of elevation (11°) and we know the length of the side adjacent to the angle of elevation. A trigonometric function that has both the opposite and adjacent sides is the tangent function. We can use this to answer the question. Empire State Building B 11° 1 mile continued on next slide

The angle of elevation to the top of the Empire State Building in New York is found to be 11 degrees from the ground at a distance of 1 mile from the base of the building. Using this information, find the height of the Empire State Building. Empire State Building B Note that this answer is in the same units as those that your single given distance is in. That distance is miles. We are asked to find the height in feet. There are 5280 feet in 1 mile. Thus the answer is .1943803091*5280=1026.328032 feet 11° 1 mile

Right Triangle Trigonometry