1)find the exact value indicated trigonometric function of θ, if θ is the angle is standard position whose terminal side contains the given point.
(-2 ,√2); Find cos θ
2) find the length of the arc intercepted by a central angle of 35degree 28' in a circle of radius 15 ft. Round to two decimal places, if necessary.
Find all trigonometric functions of angle in standard position whose terminal arm passes
through the point a) (−√3, 1 ). b) (-3, -4 ).
1) To find the value of cos θ, we can use the given point (-2, √2). We start by drawing a right triangle in the coordinate plane with one side along the x-axis and another side along the y-axis. The hypotenuse of the triangle is the distance between the origin and the given point.
Since the x-coordinate of the point is -2 and the y-coordinate is √2, we have a right triangle with legs of length 2 and √2. The hypotenuse can be found using the Pythagorean theorem:
hypotenuse^2 = leg1^2 + leg2^2
hypotenuse^2 = 2^2 + (√2)^2
hypotenuse^2 = 4 + 2
hypotenuse^2 = 6
hypotenuse = √6
Now, we can use the values of the legs and the hypotenuse to find the trigonometric function cos θ:
cos θ = adjacent/hypotenuse
cos θ = 2/√6
To rationalize the denominator, we multiply both the numerator and the denominator by √6:
cos θ = (2/√6) * (√6/√6)
cos θ = 2√6 / 6
cos θ = √6 / 3
Therefore, the exact value of cos θ for the given point (-2, √2) is √6 / 3.
2) To find the length of the arc intercepted by a central angle of 35 degrees 28 minutes (35° 28'), we can use the formula:
Arc length = (central angle / 360°) * (circumference of the circle)
First, we need to convert the central angle from degrees and minutes to decimal degrees. There are 60 minutes in one degree, so 28 minutes is equal to 28/60 = 0.47 degrees.
Now, we can calculate the arc length:
Arc length = (35 + 0.47) / 360 * (2π * radius)
Arc length = 35.47 / 360 * (2π * 15)
Arc length ≈ 0.098 * (2 * 3.14 * 15)
Arc length ≈ 0.098 * (6.28 * 15)
Arc length ≈ 0.098 * 94.2
Arc length ≈ 9.23 (rounded to two decimal places)
Therefore, the length of the arc intercepted by a central angle of 35° 28' in a circle of radius 15 ft is approximately 9.23 ft.