Find the value of the six trigonometric functions.
t = 7(pi)/4
Can you show me step by step so I can see how this is solved? Also, do I need to graph this first in order to understand or begin the problem? I am trying to self teach - but having problems.
t = 7(pi)/4 Radians.
7(p1)/4 * 180/(pi) = 315 deg,
315 deg. is in 4th quadrant where sine
and tangent are negative.
sin315 = -0.707 = -sqrt(2)/2 = y/r,
x^2 + y^2 = r^2,
x^2 + (-sqrt2)^2 = 2^2,
x^2 + 2 = 4,
x^2 = 2,
x = sqrt(2).
sin315 = y/r = -sqrt(2)/2,
cos315 = x/r = sqrt(2)/2,
tan315 = y/x = -sqrt(2)/sqrt(2) = -1,
csc315 = 1/sin315 = 2/sqrt(2) = 2sqrt2/2 = sqrt2.
sec315 = 1/cos315 = 2/swqrt2 = 2sqrt2/2
= sqrt2.
cot315 = 1/tan315 = 1/-1 = -1.
Notice that no radicals were left in the denominators.
The 6 trig functions in decimal form
are easily found using the calculator:
sin315 = -0.707,
cos315 = 0.707,
tan315 = -1,
csc315 = 1/sin315 = -1.414,
sec315 = 1/cos315 = 1.414,
cot315= 1/tan315 = -1.
To find the values of the six trigonometric functions at t = 7π/4, you do not necessarily need to graph it. Graphing can be helpful to visualize the angle, but it is not required in order to find the trigonometric function values.
Let's break down the process step by step:
Step 1: Determine the reference angle.
The reference angle is the acute angle between the terminal side of the angle and the x-axis. To find the reference angle, subtract the nearest multiple of π (180 degrees) from the given angle. In this case, the nearest multiple of π is π, so subtracting π from 7π/4 gives you 7π/4 - π = 3π/4.
Step 2: Determine the quadrant.
Since 7π/4 is in the third quadrant (180 to 270 degrees), we know that the values of sine and cosecant will be negative, while the values of cosine and secant will be positive. Tangent and cotangent can be positive or negative depending on the quadrant.
Step 3: Find the trigonometric function values.
We can use the reference angle to find the trigonometric function values.
Sine (sin): To find sin(t), we need to find the y-coordinate of the point on the unit circle that corresponds to the reference angle. The unit circle is a circle with a radius of 1 centered at the origin. For the reference angle of 3π/4, the y-coordinate is (√2)/2, which is positive because the angle is in the third quadrant.
Cosine (cos): To find cos(t), we need to find the x-coordinate of the point on the unit circle that corresponds to the reference angle. For the reference angle of 3π/4, the x-coordinate is - (√2)/2, which is negative because the angle is in the third quadrant.
Tangent (tan): To find tan(t), divide the y-coordinate (sin(t)) by the x-coordinate (cos(t)). For the reference angle of 3π/4, tan(t) = sin(t) / cos(t) = (√2)/2 / (-√2)/2 = -1, which is negative because the angle is in the third quadrant.
Cosecant (csc): To find csc(t), take the reciprocal of sine. csc(t) = 1 / sin(t) = 1 / ((√2)/2) = 2 / √2 = √2.
Secant (sec): To find sec(t), take the reciprocal of cosine. sec(t) = 1 / cos(t) = 1 / (-√2)/2 = -2 / √2 = -√2.
Cotangent (cot): To find cot(t), take the reciprocal of tangent. cot(t) = 1 / tan(t) = 1 / (-1) = -1.
So, the values of the six trigonometric functions at t = 7π/4 are:
sin(t) = (√2)/2
cos(t) = - (√2)/2
tan(t) = -1
csc(t) = √2
sec(t) = -√2
cot(t) = -1