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Test your knowledge
Textbook: pp. 100–102
GoalReview Chapter II (sine, cosine, tangent, cotangent; laws of sines and cosines; area; dot product) and prepare for test questions.
New words
law of sines · sinuslar teoremasilaw of cosines · kosinuslar teoremasidot product · skalar ko‘paytmaHeron's formula · Geron formulasi
Explanation
A short reminder. sin²α + cos²α = 1; sin(180° − α) = sinα; cos(180° − α) = −cosα. The area of a triangle is S = ½ab·sinC. Law of sines: a/sinA = b/sinB = c/sinC = 2R. Law of cosines: a² = b² + c² − 2bc·cosA. With three sides the area comes from Heron's formula, and R = abc/(4S). Dot product a·b = |a||b|cosφ = a₁b₁ + a₂b₂, and it is 0 for perpendicular vectors. In test questions choose the theorem from what is given: “two angles and a side” — sines; “two sides and the included angle” or “three sides” — cosines. Know the values for the special angles 30°, 45°, 60°, 90°, 120°, 135°, 150° by heart.
Worked examples
Test: in a triangle with sides 5, 5, 8, cos(angle opposite 8) = (25 + 25 − 64)/50 = −14/50 = −7/25, so the angle is obtuse.
Test: a(2; 5) and b(5; −2): 10 − 10 = 0, so they are perpendicular.
Class activity
“Test race”: 10 questions in 15 minutes; swap and check in pairs, noting the type of error (wrong formula, sign, special value).
Practice
1
Find cos135° + sin135°.
0
2
A triangle has sides 6 and 8 with a 90° angle between them. What is its area?
24
3
a = 4 and ∠A = 30°. What is the circumradius?
4
4
When is the law of sines convenient and when the law of cosines? Give an example.
Sines: two angles and a side are given. Cosines: two sides and the included angle, or three sides.