Finally, in quadrant IV, “ Calculus” only cosine and its reciprocal function, secant, are positive. In quadrant III, “ Take,” only tangent and its reciprocal function, cotangent, are positive. In quadrant II, “ Students,” only sine and its reciprocal function, cosecant, are positive. This course does not provide sufficient background for students who will need to take Precalculus Algebra or Calculus. In quadrant I, which is “ A,” all of the six trigonometric functions are positive. In the fourth quadrant (270 to 360), cos and sec are positive and other trigonometric ratios are negative. To help us remember which of the six trigonometric functions are positive in each quadrant, we can use the mnemonic phrase “All Students Take Calculus” Each of the four words in the phrase corresponds to one of the four quadrants, starting with quadrant I and rotating counterclockwise. Figure 3 shows which functions are positive in which quadrant. The trigonometric function values for the original angle will be the same as those for the reference angle, except for the positive or negative sign, which is determined by x– and y-values in the original quadrant. The sign depends on the quadrant of the original angle. We can find the exact trig value of any angle in any quadrant if we apply the trig function to the reference angle. We will use the reference angle of the angle of rotation combined with the quadrant in which the terminal side of the angle lies. They can also be used to find \left(x,y\right) coordinates for those angles. ACTS still starts in quadrant 1 but goes clockwise going through quadrants 1, 4, 3, then 2.Reference angles make it possible to evaluate trigonometric functions for angles outside the first quadrant.CAST still goes counterclockwise but starts in quadrant 4 going through quadrants 4, 1, 2, then 3.These have the disadvantages of not going sequentially from quadrants 1 to 4 and not reinforcing the numbering convention of the quadrants. Other easy-to-remember mnemonics are the ACTS and CAST laws. Quadrant IV (angles from 270 to 360 degrees, or 3π/2 to 2π radians): Cosine and secant functions are positive in this quadrant.Quadrant III (angles from 180 to 270 degrees, or π to 3π/2 radians): Tangent and cotangent functions are positive in this quadrant.Quadrant II (angles from 90 to 180 degrees, or π/2 to π radians): Sine and cosecant functions are positive in this quadrant.Quadrant I (angles from 0 to 90 degrees, or 0 to π/2 radians): All trigonometric functions are positive in this quadrant.The letters ASTC signify which of the trigonometric functions are positive, starting in the top right 1st quadrant and moving counterclockwise through quadrants 2 to 4. Signs of trigonometric functions in each quadrant.Īll Students Take Calculus is a mnemonic for the sign of each trigonometric functions in each quadrant of the plane. Longer mnemonics for these letters include "Oscar Has A Hold On Angie" and "Oscar Had A Heap of Apples." All Students Take Calculus Īn alternate way to remember the letters for Sin, Cos, and Tan is to memorize the nonsense syllables Oh, Ah, Oh-Ah (i.e. Communities in Chinese circles may choose to remember it as TOA-CAH-SOH, which also means 'big-footed woman' ( Chinese: 大腳嫂 Pe̍h-ōe-jī: tōa-kha-só) in Hokkien. The order may be switched, as in "Tommy On A Ship Of His Caught A Herring" (tangent, sine, cosine) or "The Old Army Colonel And His Son Often Hiccup" (tangent, cosine, sine) or "Come And Have Some Oranges Help To Overcome Amnesia" (cosine, sine, tangent). Phrases Īnother method is to expand the letters into a sentence, such as "Some Old Horses Chew Apples Happily Throughout Old Age", "Some Old Hippy Caught Another Hippy Tripping On Acid", or "Studying Our Homework Can Always Help To Obtain Achievement". From the saying All Students Take Calculus determine that the sine is positive in this. ˌ s oʊ k ə ˈ t oʊ ə/ SOH-kə- TOH-ə, similar to Krakatoa). Study for pre-calculus with our comprehensive study guide. One way to remember the letters is to sound them out phonetically (i.e. Follow the All Students Take Calculus mnemonic rule (ASTC) to remember when these functions are positive. Sine = Opposite ÷ Hypotenuse Cosine = Adjacent ÷ Hypotenuse Tangent = Opposite ÷ Adjacent positive and negative in various quadrants. When the reference angle comes out to be 0, 30, 45, 60, or 90 degrees, you can use the function value of that angle and then figure out the sign of the angle. The sine, cosine, and tangent ratios in a right triangle can be remembered by representing them as strings of letters, for instance SOH-CAH-TOA in English: The first step to finding the trig function value of one of the angles that’s a multiple of 30 or 45 degrees is to find the reference angle in the unit circle. Image mnemonic to help remember the ratios of sides of a right triangle
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