22
Relations between the sine, cosine and tangent of the same angle
Textbook: pp. 112–116
GoalKnow the basic relations among sine, cosine, tangent and cotangent of one angle and use them to compute values.
New words
fundamental trigonometric identity · asosiy trigonometrik ayniyatrelation of tangent and cotangent · tangens va kotangensning bog‘lanishisign before the root · ildiz oldidagi ishoraequation of the unit circle · birlik aylana tenglamasi
Explanation
For the point M(cos α, sin α) of the unit circle we have x² + y² = 1, so sin²α + cos²α = 1, the fundamental trigonometric identity, true for every α. From it sin α = ±√(1 − cos²α) and cos α = ±√(1 − sin²α); the sign before the root is chosen from the sign in the quadrant where the angle lies. From the definitions, tan α · cot α = 1 (α ≠ πk/2), i.e. cot α = 1 : tan α. Dividing the fundamental identity by cos²α gives 1 + tan²α = 1 : cos²α (cos α ≠ 0). With these formulas, if one function value and the quadrant are known, we can find the others.
Worked examples
sin α = 5/13, π/2 < α < π. In quadrant II cos α < 0: cos α = −√(1 − 25/169) = −√(144/169) = −12/13. tan α = sin α : cos α = −5/12.
tan α = 2, π < α < 3π/2. 1 + tan²α = 5 = 1 : cos²α, so cos²α = 1/5. In quadrant III cos α < 0: cos α = −√5/5. sin α = tan α · cos α = −2√5/5.
Class activity
“Triangle helper”: a right triangle (3, 4, 5) is drawn; students write sin, cos, tan as ratios of sides and check sin²α + cos²α = 1, then adjust the signs for quadrant II.
Practice
1
If sin α = 3/5 and 0 < α < π/2, find cos α and tan α.
cos α = 4/5, tan α = 3/4
2
If cos α = −5/13 and π/2 < α < π, find sin α.
12/13
3
If tan α = 1/4, what is cot α?
4
4
Why is sin²α + cos²α = 1?
The point (cos α, sin α) lies on the unit circle, whose equation is x² + y² = 1.