Cos 150 value

\( \newcommand\) • • • • • • • • • • • • • • • • Learning Objectives • Find function values for the sine and cosine of 30° or \((\frac\): The Singapore Flyer is the world’s tallest Ferris wheel. (credit: “Vibin JK”/Flickr) 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 Figure \(\PageIndex\): Unit circle where the central angle is \(t\) radians UNIT CIRCLE A unit circle has a center at \((0,0)\) and radius \(1\). The length of the intercepted arc is equal to the radian measure of the central angle \(t\). Let \((x,y)\) be the endpoint on the unit circle of an arc of arc length \(s\). The \((x,y)\) coordinates of this point can be described as functions of the angle. Defining Sine and Cosine Functions Now that we have our unit circle labeled, we can learn how the \((x,y)\) coordinates relate to the arc length and angle. The sine function relates a real number \(t\) to the \(y\)-coordinate of the point where the corresponding angle intercepts the unit circle. More precisely, the sine of an angle \(t\) equals the \(y\)-value of the endpoint on the unit circle of an arc of length \(t\). In Figure \(\PageIndex\) Because it is understood that sine and cosine are functions, we do not always need to write them with parentheses: \(\sin t\) is the same as \(\sin (t)\) and \(\cos t\) is the same as \(\cos (t)\). Likewise, \(\cos ^2 t\) is a commonly used...

First let's calculate #sin(-150)# #sin(-150)=-sin150# we can write this, because #sinx# is an odd function. #-sin150=-sin(180-30)=-sin30=-1/2# Now we can calculate #cos(-150)# using: #sin^2x+cos^2x=1# #(-1/2)^2+cos^2(-150)=1# #cos^2(-150)=1-1/4# #cos^2(-150)=3/4# #cos(-150)=-sqrt(3)/2# We choose the negative value because the end arm of the angle lies in the third quadrant, and in this quadrant #sin# and #cos# are negative. Tio calculate #tan(-150)# using: #tanx=sinx/cosx# Here it is: #tan(-150)=sin(-150)/cos(-150)=-1/2-:(-sqrt(3)/2)=# #=1/2xx2/sqrt(3)=1/sqrt(3)=sqrt(3)/3#

First method Trig table, unit circle, and property of complementary arcs --> #cos 150 = cos (60 + 90) = - sin 60 = - sqrt3/2# Second method: Use trig identity: cos (a + b) = cos a.cos b - sin a.sin b cos (150) = cos (60 + 90) = cos 60.cos 90 - sin 60.sin 90 = = - sin 60 = # -sqrt3/2# Note. #cos 90^@ = 0#, and #sin 90^@ = 1#

home / trigonometry / trigonometric functions / cosine Cosine Cosine, written as cos⁡(θ), is one of the six fundamental Cosine definitions There are two main ways in which trigonometric functions are typically discussed: in terms of Right triangle definition For a right triangle with an acute angle, θ, the sine value of this angle is defined to be the ratio of the adjacent side length to the hypotenuse length. The sides of the right triangle are referenced as follows: • Adjacent: the side next to θ that is not the hypotenuse • Opposite: the side opposite θ. • Hypotenuse: the longest side of the triangle opposite the right angle. Example: A plane is on a flying over a person. The person records an angle of elevation of 25° when the straight-line distance (hypotenuse of the triangle) between the person and the plane is 14 miles. What is the horizontal distance between the plane and the person? Given the information above, we can form a right triangle such that x is the horizontal distance between the person and the plane, the straight-line distance between the person and the plane is the hypotenuse, and the vertical distance between the terminal ends of x and the hypotenuse forms the right angle of the triangle. We can then find the horizontal distance, x, using the cosine function: x = 14 × cos⁡(25°) ≈ 12.69 The horizontal distance between the person and the plane is about 12.69 miles. Unit circle definition Trigonometric functions can also be defined as coordinate values o...

home / trigonometry / trigonometric functions / cosine Cosine Cosine, written as cos⁡(θ), is one of the six fundamental Cosine definitions There are two main ways in which trigonometric functions are typically discussed: in terms of Right triangle definition For a right triangle with an acute angle, θ, the sine value of this angle is defined to be the ratio of the adjacent side length to the hypotenuse length. The sides of the right triangle are referenced as follows: • Adjacent: the side next to θ that is not the hypotenuse • Opposite: the side opposite θ. • Hypotenuse: the longest side of the triangle opposite the right angle. Example: A plane is on a flying over a person. The person records an angle of elevation of 25° when the straight-line distance (hypotenuse of the triangle) between the person and the plane is 14 miles. What is the horizontal distance between the plane and the person? Given the information above, we can form a right triangle such that x is the horizontal distance between the person and the plane, the straight-line distance between the person and the plane is the hypotenuse, and the vertical distance between the terminal ends of x and the hypotenuse forms the right angle of the triangle. We can then find the horizontal distance, x, using the cosine function: x = 14 × cos⁡(25°) ≈ 12.69 The horizontal distance between the person and the plane is about 12.69 miles. Unit circle definition Trigonometric functions can also be defined as coordinate values o...

First method Trig table, unit circle, and property of complementary arcs --> #cos 150 = cos (60 + 90) = - sin 60 = - sqrt3/2# Second method: Use trig identity: cos (a + b) = cos a.cos b - sin a.sin b cos (150) = cos (60 + 90) = cos 60.cos 90 - sin 60.sin 90 = = - sin 60 = # -sqrt3/2# Note. #cos 90^@ = 0#, and #sin 90^@ = 1#

First let's calculate #sin(-150)# #sin(-150)=-sin150# we can write this, because #sinx# is an odd function. #-sin150=-sin(180-30)=-sin30=-1/2# Now we can calculate #cos(-150)# using: #sin^2x+cos^2x=1# #(-1/2)^2+cos^2(-150)=1# #cos^2(-150)=1-1/4# #cos^2(-150)=3/4# #cos(-150)=-sqrt(3)/2# We choose the negative value because the end arm of the angle lies in the third quadrant, and in this quadrant #sin# and #cos# are negative. Tio calculate #tan(-150)# using: #tanx=sinx/cosx# Here it is: #tan(-150)=sin(-150)/cos(-150)=-1/2-:(-sqrt(3)/2)=# #=1/2xx2/sqrt(3)=1/sqrt(3)=sqrt(3)/3#

\( \newcommand\) • • • • • • • • • • • • • • • • Learning Objectives • Find function values for the sine and cosine of 30° or \((\frac\): The Singapore Flyer is the world’s tallest Ferris wheel. (credit: “Vibin JK”/Flickr) 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 Figure \(\PageIndex\): Unit circle where the central angle is \(t\) radians UNIT CIRCLE A unit circle has a center at \((0,0)\) and radius \(1\). The length of the intercepted arc is equal to the radian measure of the central angle \(t\). Let \((x,y)\) be the endpoint on the unit circle of an arc of arc length \(s\). The \((x,y)\) coordinates of this point can be described as functions of the angle. Defining Sine and Cosine Functions Now that we have our unit circle labeled, we can learn how the \((x,y)\) coordinates relate to the arc length and angle. The sine function relates a real number \(t\) to the \(y\)-coordinate of the point where the corresponding angle intercepts the unit circle. More precisely, the sine of an angle \(t\) equals the \(y\)-value of the endpoint on the unit circle of an arc of length \(t\). In Figure \(\PageIndex\) Because it is understood that sine and cosine are functions, we do not always need to write them with parentheses: \(\sin t\) is the same as \(\sin (t)\) and \(\cos t\) is the same as \(\cos (t)\). Likewise, \(\cos ^2 t\) is a commonly used...