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Thought-Process to Discover Knowledge

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jogger,

exercise.

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exercise provides practice problems to become fluent in the concepts.

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summary of this topic

### Trigonometric Ratio for Angles in All Quadrants

Voice

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Home

»  90+theta
→  theta measured with y axis
→  complementary functions

»  180-theta
→  theta measured with x axis
→  equivalent functions

plain and simple summary

nub

plain and simple summary

nub

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simple steps to build the foundation

trek

simple steps to build the foundation

trek

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The results of representing trigonometric ratios of angles in the 2nd quadrant equivalently as trigonometric ratios of acute angle in the 1st quadrant are explained.

Keep tapping on the content to continue learning.
Starting on learning representing trigonometric ratios for "Angles in 2nd Quadrant". ;; The results of representing trigonometric ratios of angles in the 2nd quadrant equivalently as trigonometric ratios of acute angle in the 1st quadrant are explained.

The angle omega = 90+theta is shown in the figure as point P. The similar triangle with angle theta is given by Q(x,y) in 1st quadrant. Note: Given Q(x,y), the coordinate of P is (color(coral)(P_x),color(deepskyblue)(P_y)):
color(coral)(P_x=-y) and color(deepskyblue)(P_y=x)

What is tan (90+theta)?

• y/x = tan theta
• (-y)/x = - tan theta
• x/y= cot theta
• x/(-y) = - cot theta

The answer is '(-x)/y = - cot theta'.

tan (90+theta) (tan of the given angle)
quad = (P_y)/(P_x) (by definition of tan)
quad = x/(-y) (substituting the values)
quad = - cot theta (equivalently in 1st quadrant.)

Note: learners can work out this for sin and cos.

For angles given as 90+theta,

•  the angle is in 2nd quadrant, so x projection is negative and y projection is positive.

•  the /_theta is measured with y-axis, and so the similar triangle in 1st quadrant will have /_theta with x-axis, thereby swapping the x and y projections.

the trigonometric ratios are:  • sin (90+theta) = cos theta

• cos (90+theta) = - sin theta

• tan (90+theta) = - cot theta

The angle omega = 180-theta is shown in the figure as point P. The similar triangle with /_theta is given by Q(x,y) in 1st quadrant. Note: Given Q(x,y), the coordinates of P is (color(coral)(P_x),color(deepskyblue)(P_y)):
color(coral)(P_x=-x) and color(deepskyblue)(P_y=y)

What is tan (180-theta)?

• y/x = tan theta
• y/(-x) = - tan theta
• x/y= cot theta
• (-x)/y = - cot theta

The answer is 'y/(-x) = - tan theta'

tan (180-theta) (tan of the given angle)
quad = (P_y)/(P_x) (by definition of tan)
quad = y/(-x) (substituting the values)
quad = - tan theta (equivalently in 1st quadrant.)

Note: learners can work out this for sin and cos.

For angles given as 180-theta, the trigonometric ratios are:  • sin (180-theta) = sin theta

• cos (180-theta) = - cos theta

• tan (180-theta) = - tan theta

comprehensive information for quick review

Jogger

comprehensive information for quick review

Jogger

dummy

practice questions to master the knowledge

Exercise

practice questions to master the knowledge

Exercise

Progress

Progress

The angle omega equals 90 plus theta is shown in the figure as point P. The similar triangle with angle theta is given by Q x y in 1st quadrant ;; Note: Given point q x, y; the coordinates of p is p x = minus y and p y = x. What is tan 90 plus theta?
tan
y by x equals tan theta
minus tan
y by minus x equals minus tan theta
cot
x by y equals cot theta
minus cot
minus x by y equals minus cot theta
The answer is 'minus x by y equals minus cot theta'
For angles given as 90 plus theta. the angle is in 2nd quadrant, so x projection is negative and y projection is positive. The angle theta is measured with y-axis, and so the similar triangle in 1st quadrant will have angle theta with x-axis, thereby swapping the x and y projections. the trigonometric ratios are: sine 90 plus theta equals cos theta;; cos 90 plus theta equals minus sine theta;; tan 90 plus theta equals minus cot theta
The angle omega equals 180 minus theta is shown in the figure as point P. The similar triangle with angle theta is given by Q x y in 1st quadrant ;; Note: Given Q x y , the coordinates of P is P x = minus x and p y = y. What is tan 180 minus theta?
equals tan
y by x equals tan theta
minus tan
y by minus x equals minus tan theta
equals cot
x by y equals cot theta
minus cot
minus x by y equals minus cot theta
The answer is ' y by minus x equals minus tan theta'
For angles given as 180 minus theta, the trigonometric ratios are: sine 180 minus theta equals sine theta;; cos 180 minus theta equals minus cos theta;; tan 180 minus theta equals minus tan theta

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