الرياضيات (بالإنجليزية) الصف الثالث الإعدادي، الأسبوع 10: The two acute angles: complementary angles and tan θ = sin θ ÷ cos θ

Trigonometric ratios of an acute angle

الأسبوع 10 · النسب المثلثية للزاوية الحادة: جيب الزاوية وجيب تمامها وظلها والزاويتان المتتامتان وأطوال الأضلاع · 3 دروس

The two acute angles: complementary angles and tan θ = sin θ ÷ cos θ

Mathematics Prep 3 Term 1, Unit 3, Lesson 4

GoalUse the two acute angles of one right triangle to link sine and cosine, and use tan θ = sin θ ÷ cos θ.

New words

complementary angles · two angles whose sum is 90°sin θ = cos β · when θ + β = 90°, the sine of one is the cosine of the othertan θ = sin θ ÷ cos θ · the tangent is the ratio of the sine to the cosinesin²θ · means (sin θ)²: the sine, squared

Explanation

The two acute angles of a right triangle add up to 90°, so they are complementary: θ + β = 90°. The side opposite θ is the side adjacent to β and the other way round. So sin θ = cos β and cos θ = sin β, and the tangents are reciprocals: tan θ × tan β = 1. In the 5-12-13 triangle with θ opposite 5: sin θ = 5/13 = cos β, cos θ = 12/13 = sin β, tan θ = 5/12 and tan β = 12/5. There is one more link. Divide the two ratios: sin θ ÷ cos θ = (opposite ÷ hyp) ÷ (adjacent ÷ hyp) = opposite ÷ adjacent. So tan θ = sin θ ÷ cos θ. Finally, by Pythagoras, sin²θ + cos²θ = 1, because (opp² + adj²) ÷ hyp² = 1.

	For θ	For β
sin	5/13	12/13
cos	12/13	5/13
tan	5/12	12/5

Worked examples

In the 8-15-17 right triangle, θ is opposite the 8. Show that tan θ = sin θ ÷ cos θ. sin θ = 8/17, cos θ = 15/17, tan θ = 8/15. (8/17) ÷ (15/17) = 8/17 × 17/15 = 8/15 ✓. For the other acute angle β: sin β = 15/17 = cos θ.

If sin θ = 3/5 find cos β where θ + β = 90°. cos β = sin θ = 3/5, since the angles are complementary. Check: in the 3-4-5 triangle with θ opposite 3, the side adjacent to β is also 3, so cos β = 3/5.

In a triangle ABC with ∠B = 90°, prove sin²A + sin²C = 1. sin A = BC ÷ AC and sin C = AB ÷ AC. So sin²A + sin²C = (BC² + AB²) ÷ AC² = AC² ÷ AC² = 1 by Pythagoras.

Class activity

Swap the angle. On a 3-4-5 triangle cut from card, mark the two acute angles θ and β. Write the six ratios (sin, cos, tan for each angle). Which pairs of ratios are equal? Which pair multiplies to 1?

Practice

In the 9-40-41 triangle θ is opposite 9. Find sin θ, and cos β for the other acute angle β.

θ + β = 90° and sin θ = 0.6. What is cos β?

tan θ = 2/3 and θ + β = 90°. Find tan β.

Check tan θ = sin θ ÷ cos θ in the 3-4-5 triangle for θ opposite 3.

In △ABC with ∠B = 90° prove tan A × tan C = 1.