Mathematics Dr. Omar Al Jadaan Assistant Professor – Computer Science & Mathematics General Education Department

Trigonometric function Trigonometry has been used for over 2 thousand years to solve many real world problems, among them surveying, navigation and problems in engineering and medical sciences. The next time you go in for an advanced scanning procedure, be sure to check out how the sine and cosine functions you learn at school find a practical application is medical techniques such as CAT and MRI scanning, in detecting tumors and even in laser treatments. The unit circle is the basis of analytic trigonometry. Angles have two sides, the initial side and the terminal side. A positive angle rotates counterclockwise, A negative angle rotates clockwise, .

Terminal side p(x,y) Initial side Angles on the unit circle are measured in radians. 2π radians = 3600 . There are six Trig functions: Sinθ=y, cosθ=x, tanθ=y/x, secθ=1/x, cscθ=1/y, cotθ=x/y Terminal side 1 p(x,y) θ Initial side

Examples 1. Find all six trigonometric functions for θ. a) The terminal point for θ is (24/25, 7/25). b) θ = π/3. sin θ = 7/25, cos θ = 24/25, csc θ =25/7 sec θ = 25/24, tan θ = 7/24, and cot θ=24/7 b) cos π/3 = ½ sin π/3 = (3)1/2 /2 sec π/3 = 2 csc = 2/ (3)1/2 tan π/3 = (3)1/2 cot π/3 = 1/ (3)1/2

Skill practice 1. Find all six trigonometric functions for θ. a) the terminal points for θ is (3/5,4/5). b) θ = π/6

Skill practice Solution: 1. a) sin θ = 4/5, cos θ = 3/5, csc θ =5/4 sec θ = 5/3, tan θ = 4/3, and cot θ=3/4 b) π/6 = 300 . So the value of sin 300 = 1/2, cos 300 = (3)1/2 /2 , csc 300 =2 sec 300 = 2/ (3)1/2, tan 300 = 1/ (3)1/2, and cot 300 = (3)1/2

Examples Sin θ = 3/6 = 1/2 . So θ = 300 . Sin 600 = h/6 . So h=3 √3 . A painter in a hospital uses a ladder of length 6 m which leans against a vertical wall so that the base of the ladder is 3 m from the wall. Calculate the angle between the ladder and the wall and also the height of the wall where it leans. Solution: 6 m h 3 m Sin θ = 3/6 = 1/2 . So θ = 300 . Sin 600 = h/6 . So h=3 √3 .

Examples Sin 300 = 50/L 1/2 = 50/L. So L = 100 m. L 50 m 300 A kite flying at a height of 50 m is attached to a string which makes an angle of 30 with the horizontal. What is the length of the string? Solution: L 50 m 300 Sin 300 = 50/L 1/2 = 50/L. So L = 100 m.

Examples 1. Find all six trigonometric ratios for θ. 5 2. A paramedic is standing 300 feet from the base of a five story building. He estimates that the angle of elevation to the top of the building is 630 . Approximately how tall is the building? 5 3 θ 4

Examples 3. Solve the triangle. c 3 30 a

Examples Solution: sin θ = 3/5, cos θ = 4/5 , tan θ = 3/4, sec θ=5/4, csc θ = 5/3 and cot θ = 4/3. We need to find b in the following triangle: tan 630 = b/300 b = 300 * 1.96 = 588 So the building is 588 feet tall. b 63 300

Example Solution: 3. sin 300 = 3/c, so ½ = 3/c, c = 6. a2 + 32 = 62 Another angle = 90 -30 = 600 .

Skill practice 1. Find all trigonometric ratios for θ. 2. A plane is flying at an altitude of 5000 feet. The angle of elevation to the plane from a car traveling on a highway is about 300 . How far apart are the plane and car? √5 1 θ 2

Skill practice 3. Solve the triangle. 4 b 60 a

More Skill practice 1. An electrician in a hospital uses a ladder of length 8 m which rests against a vertical wall so that the angle between the ladder and the wall is 300. How far is the base of the ladder from the wall. Find the height of the wall where the ladder leans. 2. A paramedic would like to know the height of the building window to rescue people from the fire and also to figure out the protective layer on the ground when people are dropped from the window. The distance from the building to the van is 5 m and angle of elevation is 300 . Find the height.

More Skill Practice 1) A surveyor is standing 50 feet from the base of a large tree. The surveyor measures the angle of elevation to the top of the tree as 71.50 . How tall is the tree? A skateboard ramp requires a rise of one foot for each three feet of horizontal length. Using the given figure, find sides b and c, and the measure of θ . 4 ft b c

More Skill practice 3. A happy little boy is flying a kite from a string of length 150 m. If the string is taut and makes an angle of 680 with the horizontal, what is the height of the kite? 150 m h 680

Sine, Cosine, and Tangent In a right-angled triangle, the size of any angle is related to the ratio of the lengths of any two sides by the trigonometric functions. The basic functions are sine, cosine, and tangent. These functions are based on the similarity of triangles that have a right angle and one other angle in common. Imagine an angle A formed by the intersection of lines AB and AC (see diagrams). A third line drawn perpendicularly up from AC gives a right-angled triangle. The sides of such a triangle will always be in the same ratio to one another, no matter where the third line intersects AC. Sine The sine of angle A is the ratio of the lengths BC to AB. Cosine The cosine of angle A is the ratio of the lengths AC to AB. Tangent The tangent of angle A is the ratio of the lengths BC to AC.

Cosine and Sine Rules Relationships between the lengths of the sides and the sizes of the angles of any triangle are given by the cosine and sine rules. Provided we have enough information already, we can use the cosine and sine rules to find the length of a side, or the size of an angle. We usually use these methods only for triangles that do not have a right angle because easier methods for finding the dimensions of right-angled triangles exist.

The Cosine Rule The cosine rule states that, for the triangle ABC in the following diagram:                 a2 = b2 + c2 - 2bcCosA

Example We can use the cosine rule to calculate the length of the third side of a triangle if we know the lengths of the other two sides and the size of the angle between them (the included angle). For example, suppose we want to find the length of side a in the triangle in following diagram                

Example Solution:                 We know that length b = 6, length c = 8, and angle A = 120°. Therefore, we can use the cosine rule, as follows:                 a2 = b2 + c2 - 2bccosA                     = 36 + 64 - 2 × 6 × 8 × cos120°                     = 36 + 64 - 96 × -0.5                     = 148 Therefore,                 a = √148                    = 12.166 (to 3 decimal places)

Example We can also use the cosine rule to find the size of any angle of a triangle if we know the lengths of all three sides. For example, suppose we want to find the size of angle A in the triangle in the following diagram:          

Example Solution:                 We know that length a = 9, length b = 6, and length c = 8. Therefore, we can use the cosine rule to find angle A, as follows:                 a2 = b2 + c2 - 2bccosA Rearranging this formula gives us                 cosA=(b2+c2 - a2) / 2bc So,                 cosA=(36+64 - 81) / 2x6x8                         =(36+64 - 81) / 96                         =19/96 = 0.19792... Therefore,                 A = cos-10.19792 = 78.585° (to 3 decimal places)

The Sine Rule The sine rule states that, for the triangle ABC shown in the following diagram:                where R is the radius of the circumcircle (the circle that passes through A, B, and C, and is said to circumscribe the triangle ABC).

Example The sine rule can be used to obtain information about a triangle if we know the length of one side and the sizes of two angles. To use the sine rule in this way, we need to know the size of the angle that is opposite the side whose length we know; thus in the triangle in Diagram we need to know the size of angle A since it is opposite the side whose length is known, a.               

Example Solution:                 The sum of the interior angles of a triangle is always 180°, so the size of angle A in Diagram is given by 180° - (B + C) = 180° - (40° + 20°) = 180° - 60° = 120°   Suppose we want to find the length of side b in the triangle in Diagram 5. We know that the length of side a = 12, that angle A = 120°, and angle B = 40° and, therefore, we can use the sine rule, as follows:        

Example Solution:                         We could find the length of side c in a similar way.

Example The sine rule can also be used to obtain information about a triangle if we know the lengths of two sides and the size of a non-included angle (that is, one of the angles that is not between the two sides whose lengths we know). If we know the lengths of two sides and the size of the included angle, we would use the cosine rule, as explained earlier. Suppose we want to find the size of angle C in the triangle in Diagram

Example Solution:                 We know that the length of side a = 10, and of side c = 4, and that angle A = 130°, and so we can use the sine rule, as follows:         Therefore,  C = sin-10.30641 = 17.843° (to 3 decimal places). We could find the size of the third angle, B, very easily by subtracting the sum of angles A and C from 180°, as we did earlier.

Skill practice 1. Find the unknown values in the following using Sine rule: a) In triangle ABC, angle A=61, angle B=47, AC=7.2. Find BC. b) In triangle ABC, Angle A=62, BC=8, AB=7. Find C. 2. Find the unknown values in the following using Cosine rule: a) In triangle ABC, AB=4 cm, AC=7 cm. Find angle A. b) In triangle ABC, angle B=117 , AB=80 cm, BC=100 cm. Find AC.

Skill practice 3. A plane is flying over a highway at an altitude of 1/2 mile. A blue car is traveling on a highway in front of the plane and a red car is on the highway behind the plane. The angle of elevation from the blue car to the plane is 300. If the cars are two miles apart, how far is the plane from each car? 4. From the top of a 200-foot lighthouse, the angle of depression to a ship on the ocean is 200. How far is the ship from the base of the lighthouse?

Skill practice Solution 3. Sin300 = 0.5/r; r = 0.5 /sin300 = 005/0.5 = 1 mile b2 = r2 + p2 – 2rp.cos300 b2 = 1 + 4 – 4 (0.86) = 5 – 3.464 = 1.535 b = √1.535 = 1.24 miles The plane is 1 mile away from red car and 1.24 mile away from the blue car. P 0.5 300 R B 2

Graph of trigonometric functions The graph of trigonometric function is a record of each cycle around the circle. For the function f(x)=sin x, x is the angle and f(x) is the y- coordinate of the terminal point determined by the angle x.

Graph of sin x function For x values select the interval -2π to 2π. y 1 -π π x -2π 2π -1

Graph of cos x function For x values select the interval -2π to 2π. y 1 -π π x -2π 2π -1

Graph of tan x function For x values select the interval -2π to 2π, if x≠ π/2, - π/2, 3π/2, -3π/2 y 1 -π x -2π π 2π -1

Inverse trigonometric function Only one-to-one function can have inverses, and the trigonometric functions are certainly not one-to-one. We can limit their domain and force them to be one-to-one. Limiting the sin function to the interval x = –π/2 to π/2 and the range is [- 1, 1]. If we limit the cos function to the interval from x = 0 to x = π, then we will have another one-to-one function. We express inverse of sin function as f(x)= arc sin(x) or sin-1 .

Inverse trigonometric Graph of inverse trigonometric functions. 1 1 π/2 π π/2 π/2 π/2 -π/2 -1 sin x cos x tan x

Example arc sin 21/2 /2 = π/4 arc tan 31/2 = π/3 arc cos (-1) = π

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