How does the related acute angle help to determine the trigonometric ratios of angles greater than 90 degrees? (in terms of the Cartesian plane)
And why is the sin, cos, and tan of the principal angle equal to the sin, cos, and tan of the related acute angle??
These aren't homework questions, I actually just don't understand these concepts. Please help me and explain it to me
Though they aren't homework questions I really am desperate to understand the answers to these questions as I have a quiz tomorrow
Always sketch the problem on your x-y coordinate system. Look at what is positive and what is negative in each quadrant.
But why is the sin, cos, and tan of the principal angle equal to the sin, cos, and tan of the related acute angle?
Huh? I am not sure what you mean or why you said that.
For example draw your angle 110 degrees
That is in quadrant 2 20 degrees left of straight up
now the tangent is -2.75 as we know. x is - and y is +
Now draw that angle of 290 degrees in quadrant 4 which is 20 degrees to the right of straight down. In fact it is the extension of the first line . x is + and y is -
the tangent is again -2.75
HOWEVER
the sin of 110 is +.94 because y is +
the sin of 290 is -.94 because y is -
If you want the angle with the same sin as 110, you must go where y is +
That is in quadrant 1
sin 70 = + .94
in quadrant 1
sin is +
cos is +
tan is +
in quadrant 2
sin is +
cos is -
tan is -
in quadrant 3
sin is -
cos is -
tan is +
in quadrant 4
sin is -
cos is +
tan is -
The related acute the angle between the ray and the x or y axis. however the sign depends on the quadrant.
I asked that because that's what it says in my notes, and I don't understand it;
But thank you for all the help you've given me
the thing is that 70 degrees, 110 degrees, 250 degrees and 290 degrees are all 20 degrees from the vertical axis and have the same absolute values of trig functions
HOWEVER the signs depend on the quadrant
Thanks! That makes total sense; I understand how to get the equivalent expressions now (:
And I'll definitely remember the signs
To understand how the related acute angle helps determine the trigonometric ratios of angles greater than 90 degrees in terms of the Cartesian plane, we need to start by understanding what an acute angle is.
An acute angle is any angle that measures less than 90 degrees. In the Cartesian plane, it is generally measured in relation to the positive x-axis. For example, if we have an angle that measures 45 degrees counterclockwise from the positive x-axis, it would be considered an acute angle.
Now, when it comes to angles greater than 90 degrees, we can convert them into a related acute angle to determine the trigonometric ratios. The related acute angle is obtained by subtracting the given angle from a multiple of 180 degrees.
Let's take an example to illustrate this. Suppose we have an angle of 150 degrees. To find the related acute angle, we subtract 150 degrees from 180 degrees (a multiple of 180 degrees): 180 - 150 = 30 degrees. So, the related acute angle for an angle of 150 degrees is 30 degrees.
Now, why is the sin, cos, and tan of the principal angle equal to the sin, cos, and tan of the related acute angle? This is because the trigonometric ratios (sin, cos, and tan) are periodic functions with a period of 360 degrees (or 2π radians). This means that the values of the trigonometric ratios repeat every 360 degrees.
When we have an angle greater than 90 degrees, we can always find an equivalent acute angle within the range of 0 to 90 degrees. This equivalent acute angle is the related acute angle we obtained earlier. Since the trigonometric ratios are periodic, the values of sin, cos, and tan for any angle will be the same as the values for its related acute angle.
For example, if we have an angle of 150 degrees, the related acute angle is 30 degrees. The sin of 150 degrees will be the same as the sin of 30 degrees. Similarly, the cos and tan of 150 degrees will be the same as the cos and tan of 30 degrees.
So, by converting angles greater than 90 degrees into their related acute angles, we can easily determine the trigonometric ratios using the values of the acute angles, which are easier to calculate and understand.