If θ is an angle in standard position and its terminal side passes through the point (-1,3), find the exact value of \sec\thetasecθ in simplest radical form.
If θ is an angle in standard position and its terminal side passes through the point (1,-3), find the exact value of \tan\thetatanθ in simplest radical form
On the axes below, make an appropriate scale and graph exactly one cycle of the trigonometric function y=3\cos 7xy=3cos7x.
Why did the angle go to the party? Because it heard there would be sin and cos!
Now, let's find the exact value of secθ. Since the terminal side passes through (-1,3), we can use the Pythagorean Theorem to find the length of the hypotenuse.
The distance from the origin (0,0) to (-1,3) is √((-1 - 0)^2 + (3 - 0)^2) = √(1 + 9) = √10.
Now, secθ is defined as the ratio of the hypotenuse to the adjacent side, so secθ = hypotenuse / adjacent side.
Since the adjacent side is -1 (since the point is in the 3rd quadrant), we have secθ = √10 / -1 = -√10.
Therefore, the exact value of secθ is -√10.
To find the exact value of secθ, we need to find the cosine of θ first.
Since the angle is in standard position and its terminal side passes through the point (-1,3), we can use the coordinates to determine the values of the trigonometric functions.
Let's calculate the length of the hypotenuse (r) using the Pythagorean theorem:
r = √((-1)^2 + 3^2) = √(1 + 9) = √10
Now, we can determine the values of sinθ, cosθ, and secθ:
cosθ = adjacent/hypotenuse = -1/√10 = -√10/10
To find secθ, we use the reciprocal of cosθ:
secθ = 1/cosθ = 1/(-√10/10) = -10/√10
To simplify the expression, we multiply both the numerator and denominator by the conjugate of the denominator (√10):
secθ = (-10/√10) * (√10/√10) = -10√10/10 = -√10
Therefore, the exact value of secθ in simplest radical form is -√10.
From the given point (-1,3) we know
x = -1, y = 3 and the point in in quadrant II
r^2 = (-1)^2 + 3^2 = 10
r = √10
secθ = r/x = -√10/1 = -√10
You must memorize all 6 trig ratios in terms of x, y, and r
e.g. tanθ = y/x
etc