Sin 150

Learning Objectives • 1.3.1 Convert angle measures between degrees and radians. • 1.3.2 Recognize the triangular and circular definitions of the basic trigonometric functions. • 1.3.3 Write the basic trigonometric identities. • 1.3.4 Identify the graphs and periods of the trigonometric functions. • 1.3.5 Describe the shift of a sine or cosine graph from the equation of the function. Trigonometric functions are used to model many phenomena, including sound waves, vibrations of strings, alternating electrical current, and the motion of pendulums. In fact, almost any repetitive, or cyclical, motion can be modeled by some combination of trigonometric functions. In this section, we define the six basic trigonometric functions and look at some of the main identities involving these functions. Radian Measure To use trigonometric functions, we first must understand how to measure the angles. Although we can use both radians and degrees, radians are a more natural measurement because they are related directly to the unit circle, a circle with radius 1. The radian measure of an angle is defined as follows. Given an angle θ , θ , let s s be the length of the corresponding arc on the unit circle ( Figure 1.30 The radian measure of an angle θ θ is the arc length s s of the associated arc on the unit circle. Since an angle of 360 ° 360 ° corresponds to the circumference of a circle, or an arc of length 2 π , 2 π , we conclude that an angle with a degree measure of 360 ° 360 ° has a radi...

Trigonometry is a branch in mathematics that relates the sides of the triangle to the angles. Using some ratios, relations, and identities for a triangle in terms of its sides and its corresponding angles, many problems(distance, height, etc) can be solved and calculated easily and intuitively. There are several standard ratios or relations between the angles and the sides of the triangle that help to solve some basic as well as complex problems. Trigonometric Ratios A Trigonometric ratio is defined as the proportion of the sides to the angles of a right-angled triangle. A standard trigonometric ratio can be obtained as the ratio of the sides to either of the acute angles of the right-angled triangles. Some definitions of some standard trigonometric ratios such as sine, cosine, and tangent are as follows, sin(θ) = opposite side / hypotenuse Cosine is the function that takes in the parameter an angle θ, which is either of the acute-angles in the right-angled triangles and is defined as the ratio of the length of the adjacent side to the hypotenuse of the right-angled triangle. In technical terms, it can be written as follow, cos(θ) = adjacent side / hypotenuse Tangent is the function that takes in the parameter an angle θ, which is either of the acute-angles in the right-angled triangles and is defined as the ratio of the length of the opposite side to the adjacent side of the right-angled triangle. In technical terms, it can be written as follows, tan(θ) = opposite side / ...

Learning Objectives In this section, you will: • Find function values for the sine and cosine of 30° or ( π 6 ) , 45° or ( π 4 ) 30° or ( π 6 ) , 45° or ( π 4 ) and 60° or ( π 3 ) . 60° or ( π 3 ) . • Identify the domain and range of sine and cosine functions. • Use reference angles to evaluate trigonometric functions. Figure 1 The Singapore Flyer was the world’s tallest Ferris wheel, until being overtaken by the High Roller in Las Vegas and the Ain Dubai in Dubai. (credit: “Vibin JK”/Flickr) Looking for a thrill? Then consider a ride on the Ain Dubai, the world's tallest Ferris wheel. Located in Dubai, the most populous city and the financial and tourism hub of the United Arab Emirates, the wheel soars to 820 feet, about 1.5 tenths of a mile. Described as an observation wheel, riders enjoy spectacular views of the Burj Khalifa (the world's tallest building) and the Palm Jumeirah (a human-made archipelago home to over 10,000 people and 20 resorts) as they travel from the ground to the peak and down again in a repeating pattern. In this section, we will examine this type of revolving motion around a circle. To do so, we need to define the type of circle first, and then place that circle on a coordinate system. Then we can discuss circular motion in terms of the coordinate pairs. Finding Function Values for the Sine and Cosine To define our trigonometric functions, we begin by drawing a unit circle, a circle centered at the origin with radius 1, as shown in t t intercepts fo...

From zero to 180 degrees, the sine graph goes from zero to a maximum 1 at 90 degrees, then back down to zero at 180 degrees. It's a symmetric hill, with height rising from minimum 0 to maximum 1, then 1 back to 0. Top of the hill is 1 at 90 degrees. 30 degrees is 60 less than 90. 150 is 60 greater than 90. 30 and 150 are opposite sides of the hill, at the same distance down from the top. Cos 30 = 1/2 Cos 150 is 1/2. Going from the origin to the top, you're half way up on the left side, then you're half way down on the right side. On a unit circle, 30 degrees is the somewhat familiar 2:1:square root of 3 right triangle, with hypotenuse 1/2 of 2, adjacent side is 1/2 of the square root of 3 and the opposite side is 1/2 of 1 = 1/2 sine of 30 is opposite side over hypotenuse or 1/2 over 1 = 1/2 On the unit circle, 150 degrees gives the same right triangle, only in the 3rd quadrant, so that sine 150 = opposite side over hypotenuse = 1/2 over 1 = 1/2. 150 degrees is 180-30 degrees. The right triangle in the 3rd quadrant is 30 degrees when you drop a vertical down from the end of the hypotenuse on the unit circle. Another way to think of this is take the sine curve and shift it left by 90 degrees. That gives the cosine curve which is an even function. In even functions. the graph is symmetric about the y axis. f(x) = f(-x) cosx = cos(-x) What had been 30 degrees is now -60 degrees, with the leftward 90 degree shift 30-90=-60 Cos(-60)=Cos(60) because it's an even function. Cos(60)...

Trigonometry is a branch in mathematics that relates the sides of the triangle to the angles. Using some ratios, relations, and identities for a triangle in terms of its sides and its corresponding angles, many problems(distance, height, etc) can be solved and calculated easily and intuitively. There are several standard ratios or relations between the angles and the sides of the triangle that help to solve some basic as well as complex problems. Trigonometric Ratios A Trigonometric ratio is defined as the proportion of the sides to the angles of a right-angled triangle. A standard trigonometric ratio can be obtained as the ratio of the sides to either of the acute angles of the right-angled triangles. Some definitions of some standard trigonometric ratios such as sine, cosine, and tangent are as follows, sin(θ) = opposite side / hypotenuse Cosine is the function that takes in the parameter an angle θ, which is either of the acute-angles in the right-angled triangles and is defined as the ratio of the length of the adjacent side to the hypotenuse of the right-angled triangle. In technical terms, it can be written as follow, cos(θ) = adjacent side / hypotenuse Tangent is the function that takes in the parameter an angle θ, which is either of the acute-angles in the right-angled triangles and is defined as the ratio of the length of the opposite side to the adjacent side of the right-angled triangle. In technical terms, it can be written as follows, tan(θ) = opposite side / ...

Calculate the value of sin 150 ° : First, determine the sign of sin 150 °. It is clear that 150 ° belongs to the second quadrant. It is known that the values of sines are positive + in the second quadrant. It is also known that, sin ( 180 - x ) ° = sin x °. Thus, sin 150 ° = sin 180 - 30 ° = sin 30 ° = 1 2. Therefore, the value of sin 150 ° is 1 2 .

From zero to 180 degrees, the sine graph goes from zero to a maximum 1 at 90 degrees, then back down to zero at 180 degrees. It's a symmetric hill, with height rising from minimum 0 to maximum 1, then 1 back to 0. Top of the hill is 1 at 90 degrees. 30 degrees is 60 less than 90. 150 is 60 greater than 90. 30 and 150 are opposite sides of the hill, at the same distance down from the top. Cos 30 = 1/2 Cos 150 is 1/2. Going from the origin to the top, you're half way up on the left side, then you're half way down on the right side. On a unit circle, 30 degrees is the somewhat familiar 2:1:square root of 3 right triangle, with hypotenuse 1/2 of 2, adjacent side is 1/2 of the square root of 3 and the opposite side is 1/2 of 1 = 1/2 sine of 30 is opposite side over hypotenuse or 1/2 over 1 = 1/2 On the unit circle, 150 degrees gives the same right triangle, only in the 3rd quadrant, so that sine 150 = opposite side over hypotenuse = 1/2 over 1 = 1/2. 150 degrees is 180-30 degrees. The right triangle in the 3rd quadrant is 30 degrees when you drop a vertical down from the end of the hypotenuse on the unit circle. Another way to think of this is take the sine curve and shift it left by 90 degrees. That gives the cosine curve which is an even function. In even functions. the graph is symmetric about the y axis. f(x) = f(-x) cosx = cos(-x) What had been 30 degrees is now -60 degrees, with the leftward 90 degree shift 30-90=-60 Cos(-60)=Cos(60) because it's an even function. Cos(60)...

Learning Objectives • 1.3.1 Convert angle measures between degrees and radians. • 1.3.2 Recognize the triangular and circular definitions of the basic trigonometric functions. • 1.3.3 Write the basic trigonometric identities. • 1.3.4 Identify the graphs and periods of the trigonometric functions. • 1.3.5 Describe the shift of a sine or cosine graph from the equation of the function. Trigonometric functions are used to model many phenomena, including sound waves, vibrations of strings, alternating electrical current, and the motion of pendulums. In fact, almost any repetitive, or cyclical, motion can be modeled by some combination of trigonometric functions. In this section, we define the six basic trigonometric functions and look at some of the main identities involving these functions. Radian Measure To use trigonometric functions, we first must understand how to measure the angles. Although we can use both radians and degrees, radians are a more natural measurement because they are related directly to the unit circle, a circle with radius 1. The radian measure of an angle is defined as follows. Given an angle θ , θ , let s s be the length of the corresponding arc on the unit circle ( Figure 1.30 The radian measure of an angle θ θ is the arc length s s of the associated arc on the unit circle. Since an angle of 360 ° 360 ° corresponds to the circumference of a circle, or an arc of length 2 π , 2 π , we conclude that an angle with a degree measure of 360 ° 360 ° has a radi...

Learning Objectives In this section, you will: • Find function values for the sine and cosine of 30° or ( π 6 ) , 45° or ( π 4 ) 30° or ( π 6 ) , 45° or ( π 4 ) and 60° or ( π 3 ) . 60° or ( π 3 ) . • Identify the domain and range of sine and cosine functions. • Use reference angles to evaluate trigonometric functions. Figure 1 The Singapore Flyer was the world’s tallest Ferris wheel, until being overtaken by the High Roller in Las Vegas and the Ain Dubai in Dubai. (credit: “Vibin JK”/Flickr) Looking for a thrill? Then consider a ride on the Ain Dubai, the world's tallest Ferris wheel. Located in Dubai, the most populous city and the financial and tourism hub of the United Arab Emirates, the wheel soars to 820 feet, about 1.5 tenths of a mile. Described as an observation wheel, riders enjoy spectacular views of the Burj Khalifa (the world's tallest building) and the Palm Jumeirah (a human-made archipelago home to over 10,000 people and 20 resorts) as they travel from the ground to the peak and down again in a repeating pattern. In this section, we will examine this type of revolving motion around a circle. To do so, we need to define the type of circle first, and then place that circle on a coordinate system. Then we can discuss circular motion in terms of the coordinate pairs. Finding Function Values for the Sine and Cosine To define our trigonometric functions, we begin by drawing a unit circle, a circle centered at the origin with radius 1, as shown in t t intercepts fo...