الأسبوع 12 · الدوال المثلثية: دائرة الوحدة وإشارتها في الأرباع ونسب الزوايا الخاصة · 3 دروس
2
The signs of the six functions in the four quadrants, and finding all the ratios from a point or from one ratio and the quadrant
Mathematics Sec 1 Term 1, Unit 4, Lesson 3
GoalDecide the sign of a ratio from the quadrant and find all the ratios from one ratio.
New words
sign of a ratio · positive or negative, decided by the signs of x and y in the quadrantreciprocal ratio · csc, sec and cot: each has the same sign as sin, cos and tan respectively
Explanation
The signs come from x (cos) and y (sin). First quadrant: x > 0 and y > 0, so all ratios are positive. Second quadrant: x < 0, y > 0, so only sin (and csc) is positive. Third quadrant: x < 0 and y < 0, so only tan (and cot) is positive, since the two negatives cancel. Fourth quadrant: x > 0 and y < 0, so only cos (and sec) is positive. Memory aid: All, Sin, Tan, Cos going anticlockwise. To find all ratios from one ratio and the quadrant: draw a right triangle with that ratio, find the third side with Pythagoras, then put the signs of the quadrant on x and y.
Quadrant
sin θ
cos θ
tan θ
First (0° to 90°)
+
+
+
Second (90° to 180°)
+
−
−
Third (180° to 270°)
−
−
+
Fourth (270° to 360°)
−
+
−
Worked examples
sin θ = 3/5 and θ is in the second quadrant. y = 3, r = 5, so |x| = 4 and x < 0: x = −4. cos θ = −4/5 and tan θ = −3/4.
cos θ = −5/13 and θ is in the third quadrant. x = −5, r = 13, |y| = 12 and y < 0, so y = −12: sin θ = −12/13 and tan θ = 12/5 (positive in the third quadrant ✓).
tan θ = −3/4 with sin θ > 0. tan is negative in quadrants 2 and 4; sin is positive in quadrant 2. So x = −4, y = 3, r = 5: sin θ = 3/5, cos θ = −4/5.
Class activity
Quadrant cards. Make four cards for the quadrants and write the signs of sin, cos, tan on each. Shuffle eight angles (such as 40°, 140°, 220°, 320°, −50°, 500°, 700°, −200°) and place each on the right card. Check one with a calculator.
Practice
1
Give the signs of sin, cos and tan for 250°.
Third quadrant: sin −, cos −, tan +.
2
cos θ = 4/5 and θ is in the fourth quadrant. Find sin θ and tan θ.
y = −3: sin θ = −3/5, tan θ = −3/4.
3
sin θ < 0 and cos θ > 0. In which quadrant is θ?
The fourth quadrant.
4
tan θ = 5/12 and cos θ < 0. Find sin θ and cos θ.
Third quadrant: x = −12, y = −5, r = 13. sin θ = −5/13, cos θ = −12/13.
