Cos 90-theta formula

Note that the image below is only for #x# in Q1 (the first quadrant). If you wish you should be able to draw it with #x# in any quadrant. Definition of #sin(x)# #(#side opposite angle #x)//(#hypotenuse #)# Definition of #cos(90^@ -x)# #(#side adjacent to angle #(90^@-x))//(#hypotenuse #)# but #(#side opposite angle #x) = (#side adjacent to angle #(90^@-x)# Therefore #sin(x) = cos(90^@ -x)# Similarly #cos(x) = sin(90^@ - x)# These can also be proven using the sine and cosine angle subtraction formulas: #cos(alpha-beta)=cos(alpha)cos(beta)+sin(alpha)sin(beta)# #sin(alpha-beta)=sin(alpha)cos(beta)-cos(alpha)sin(beta)# Applying the former equation to #cos(90^@-x)#, we see that #cos(90^@-x)=cos(90^@)cos(x)+sin(90^@)sin(x)# #cos(90^@-x)=0*cos(x)+1*sin(x)# #cos(90^@-x)=sin(x)# Applying the latter to #sin(90^@-x)#, we can also prove that #sin(90^@-x)=sin(90^@)cos(x)-cos(90^@)sin(x)# #sin(90^@-x)=1*cos(x)-0*sin(x)# #sin(90^@-x)=cos(x)#

Trigonometric ratios of 90 degree minus theta is one of the branches of ASTC formula in trigonometry. Trigonometric-ratios of 90 degree minus theta are given below. sin (90 ° - θ) = cos θ cos (90 ° - θ) = sin θ tan (90 ° - θ) = cot θ csc (90 ° - θ) = sec θ sec (90 ° - θ) = csc θ cot (90 ° - θ) = tan θ Let us see, how the trigonometric ratios of 9 0 degree minus theta are determined. To know that, first we have to understand ASTC formula. The ASTC formula can be remembered easily using the following phrases. "All Sliver Tea Cups" or "All Students Take Calculus" ASTC formula has been explained clearly in the figure given below. From the above picture, it is very clear that (90° -θ) falls in the first quadrant In the first quadrant (90° - θ) , all trigonometric ratios are positive. Important Conversions When we have the angles 90 ° and 270 ° in the trigonometric ratios in the form of (9 0 ° + θ) (9 0 ° - θ) (27 0 ° + θ) (270 ° -θ) We have to do the following conversions, sin θ cos θ tan θ cot θ csc θ sec θ For example, sin (270 ° + θ) = - cos θ cos (90 ° - θ) = sin θ For the angles 0 ° or 360 ° and 180 °, we should not make the above conversions. Evaluation of Trigonometric Ratios 90 Degree Minus Theta Problem 1 : Evaluate : sin (9 0 ° - θ) Solution : To evaluate sin (9 0 ° - θ), we have to consider the following important points. (i) (90 ° - θ) will fall in the I st quadrant. (ii) When we have 9 0 °, "sin" will become "cos". (iii) In the I st quadrant, the sign of "sin" i...

You can ONLY use the Pythagorean Theorem when dealing with a right triangle. The law of cosines allows us to find angle (or side length) measurements for triangles other than right triangles. The third side in the example given would ONLY = 15 if the angle between the two sides was 90 degrees. In the example in the video, the angle between the two sides is NOT 90 degrees; it's 87. As such, that opposite side length isn't 15; it's 14.6. Good question! In science at least, here is the difference between a theory/theorem and a law: A theory is an explanation for a natural occurrence. It tells the "why" about something, but it has not necessarily been proven. A law, on the other hand, states a fact- something that always happens. It tells the "what" without explaining why, and it should always be true. - [Voiceover] Let's say that I've got a triangle, and this side has length b, which is equal to 12, 12 units or whatever units of measurement we're using. Let's say that this side right over here, this side right over here, has length c, and that happens to be equal to nine. And that we want to figure out the length of this side, and this side has length a, so we need to figure out what a is going to be equal to. Now, we won't be able to figure this out unless we also know the angle here, because you could bring the blue side and the green side close together, and then a would be small, but if this angle was larger than a would be larger. So we need to know what this angle is as...

Trigonometric ratios of 90 degree minus theta is one of the branches of ASTC formula in trigonometry. Trigonometric-ratios of 90 degree minus theta are given below. sin (90 ° - θ) = cos θ cos (90 ° - θ) = sin θ tan (90 ° - θ) = cot θ csc (90 ° - θ) = sec θ sec (90 ° - θ) = csc θ cot (90 ° - θ) = tan θ Let us see, how the trigonometric ratios of 9 0 degree minus theta are determined. To know that, first we have to understand ASTC formula. The ASTC formula can be remembered easily using the following phrases. "All Sliver Tea Cups" or "All Students Take Calculus" ASTC formula has been explained clearly in the figure given below. From the above picture, it is very clear that (90° -θ) falls in the first quadrant In the first quadrant (90° - θ) , all trigonometric ratios are positive. Important Conversions When we have the angles 90 ° and 270 ° in the trigonometric ratios in the form of (9 0 ° + θ) (9 0 ° - θ) (27 0 ° + θ) (270 ° -θ) We have to do the following conversions, sin θ cos θ tan θ cot θ csc θ sec θ For example, sin (270 ° + θ) = - cos θ cos (90 ° - θ) = sin θ For the angles 0 ° or 360 ° and 180 °, we should not make the above conversions. Evaluation of Trigonometric Ratios 90 Degree Minus Theta Problem 1 : Evaluate : sin (9 0 ° - θ) Solution : To evaluate sin (9 0 ° - θ), we have to consider the following important points. (i) (90 ° - θ) will fall in the I st quadrant. (ii) When we have 9 0 °, "sin" will become "cos". (iii) In the I st quadrant, the sign of "sin" i...

You can ONLY use the Pythagorean Theorem when dealing with a right triangle. The law of cosines allows us to find angle (or side length) measurements for triangles other than right triangles. The third side in the example given would ONLY = 15 if the angle between the two sides was 90 degrees. In the example in the video, the angle between the two sides is NOT 90 degrees; it's 87. As such, that opposite side length isn't 15; it's 14.6. Good question! In science at least, here is the difference between a theory/theorem and a law: A theory is an explanation for a natural occurrence. It tells the "why" about something, but it has not necessarily been proven. A law, on the other hand, states a fact- something that always happens. It tells the "what" without explaining why, and it should always be true. - [Voiceover] Let's say that I've got a triangle, and this side has length b, which is equal to 12, 12 units or whatever units of measurement we're using. Let's say that this side right over here, this side right over here, has length c, and that happens to be equal to nine. And that we want to figure out the length of this side, and this side has length a, so we need to figure out what a is going to be equal to. Now, we won't be able to figure this out unless we also know the angle here, because you could bring the blue side and the green side close together, and then a would be small, but if this angle was larger than a would be larger. So we need to know what this angle is as...

Note that the image below is only for #x# in Q1 (the first quadrant). If you wish you should be able to draw it with #x# in any quadrant. Definition of #sin(x)# #(#side opposite angle #x)//(#hypotenuse #)# Definition of #cos(90^@ -x)# #(#side adjacent to angle #(90^@-x))//(#hypotenuse #)# but #(#side opposite angle #x) = (#side adjacent to angle #(90^@-x)# Therefore #sin(x) = cos(90^@ -x)# Similarly #cos(x) = sin(90^@ - x)# These can also be proven using the sine and cosine angle subtraction formulas: #cos(alpha-beta)=cos(alpha)cos(beta)+sin(alpha)sin(beta)# #sin(alpha-beta)=sin(alpha)cos(beta)-cos(alpha)sin(beta)# Applying the former equation to #cos(90^@-x)#, we see that #cos(90^@-x)=cos(90^@)cos(x)+sin(90^@)sin(x)# #cos(90^@-x)=0*cos(x)+1*sin(x)# #cos(90^@-x)=sin(x)# Applying the latter to #sin(90^@-x)#, we can also prove that #sin(90^@-x)=sin(90^@)cos(x)-cos(90^@)sin(x)# #sin(90^@-x)=1*cos(x)-0*sin(x)# #sin(90^@-x)=cos(x)#