Use the given information to determine the remaining five trigonometric values. cosA=-3/4 90 degrees < A < 180 degrees
The domain is restricted to values of A between 90 degrees (pi/2) and 180 degrees (pi), which will necessarily yield results in quadrant II (negative x values, positive y values). We can use the Pythagorean trig identity; sin^2A + cos^2A = 1 sin^2A = 1 - cos^2A sin^2A = 1 - (-3/4)^2 = 7/16 √(sin^2A) = √(7/16) <---remember, this value must be positive, as our angle is in quadrant II sinA = (√7)/4 tanA = sinA/cosA = -(√7)/3 secA = 1/cosA = -4/3 cscA = 1/sinA = 4/√7 -->(rationalize)--> 4(√7)/7 cotA = cosA/sinA = -3(√7)/7
Find the six trigonometric identities of the angle A=45 degrees on the unit circle.
First of all, we need to convert our angle from degrees to radians. We do so by multiplying the degree value by pi/180. A * (pi/180) = 45pi/180 = pi/4 Now, most people will tell you that you need to memorize the unit circle. You do. However, that task is made much easier if you can understand how we get these values and learn how to calculate them yourself. The unit circle is a circle in standard position with a radius r=1. Each angle touches the circle at a certain point on its circumference, (x,y). The trigonometric identities of an angle are defined as follows; sinA = y/r cosA = x/r tanA = sinA/cosA = y/x secA = 1/cosA = r/x cscA = 1/sinA = r/y cotA = cosA/sinA = x/y When given an angle as in this problem, we can calculate the trig identities by constructing a right triangle in standard position with central angle A whose hypotenuse is the radius of our circle (in this case, r=1). The length of each remaining side will correspond to the x and y values of the point at which the angle intersects the arc, and from there we can find our trig identities. We know that a right triangle with an angle of 45 degrees must be isosceles, and therefor have two sides of equal measure. We know one side, the hypotenuse, to be r=1. Using the Pythagorean Theorem, c^2=a^2+b^2; since a=b in an isosceles triangle; 1^2=2(x^2) x^2=1/2 √x=√(1/2) Rationalize: x=(√2)/2 =y sinA = (√2)/2 cosA = (√2)/2 tanA = sinA/cosA = ((√2)/2)/((√2)/2) = 1 secA = 1/cosA = √2 cscA = 1/sinA = √2 cotA = ((√2)/2)/((√2)/2) = 1
There is one point at (0,1), and another point at (3,7). They are connected by a line. Use the formula for slope to calculate the distance between these two points.
The formula for a line's slope (or the distance between two points), is defined as: m = (y2-y1)/(x2-x1). In this problem, (x1,y1) = (0,1) and (x2,y2) = (3,7) So m= (7-1)/(3-0) Simplify: m = 6/3 = 2 The distance between the two points is two units.