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Carpentry & Joinery Phase 4 Trigonometry Trigonometry Tangent Ө adjacen t cosine sine Right Angle Sample Questions & Solutions Sample Questions & Solutions
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TrigonometryTrigonometry
Tangent
Өadjacent
cosine
sine
Right
Ang
le
Sample Questions & SolutionsSample Questions & Solutions
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Pythagoras TheoremPythagoras Theorem• Given a = 3.25m & b = 3.45m calculate
the length of c
• a² + b² = c²
• 3.25² + 3.45² = c²
• 10.5635 + 11.9025 = c²
• 22.466 = c²
• c = √22.466
• c = 4.74m Answer 4.74m Answer
ca
b
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QuestionQuestionCopy each of the triangles below and label each of the sides and the angle.
A
(a)
(c)
A
(b)
A
(d)
A
(e)A
(f)
A
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AnswerAnswerCopy each of the triangles below and label each of the sides and the angle.
A
(a)
(c)
A
(b)
A
(d)
A
(e)A
(f)
A
hyphyp
adjadj
oppopp
hyphyp
hyphyp
hyphyp
hyphyp
hyphyp
oppoppoppopp
oppopp
oppopp
adjadj
adjadj
adjadj
adjadj
adjadj
oppopp
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QuestionQuestion• Use your calculator to answer to 4 decimal
places (a) cos 25° (b) sin 50° (c) tan 34°
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AnswerAnswer• Use your calculator to answer to 4 decimal
places (a) cos 25° (b) sin 50° (c) tan 34°
• (a) cos 25 = 0.906307787
= 0.90630.9063
• (b) sin 50 = 0.766044443
= 0.76600.7660
• (c) tan 34 = 0.674508516
= 0.67450.6745
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QuestionQuestion• Use your calculator to find the angles to the
nearest degree
(a) sin A = 0.83 (b) cos B = ¾ (c) tan A = 5/7
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AnswerAnswer• Use your calculator to find the angles to the
nearest degree
(a) sin A = 0.83 (b) cos B = ¾ (c) tan A = 5/7
• (a) 2ndF, sin, .83 = 56.098738 = 56°56°
• (b) 2ndF, cos, a/b, 3 ↓ 4 = 41.40962211 = 41°41°
• (c) 2ndF, tan, a/b, 5 ↓ 7 = 35.53767779 = 36°36°
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Solving Right-angle TrianglesSolving Right-angle Triangles• In the triangle below, find the length of b
• The side we must find is the adjacent (b)
• We have the hypotenuse c = 10m
• The ratio that uses adj. & hyp. is cos
• So cos 60° = =
• 0.5 =
• b = 10(0.5)
• b = 5m Answer
c = 10m
a
b60°
adjacenthypotenuse
b10
b10
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Solving Right-angle TrianglesSolving Right-angle Triangles• Find the angle A to the nearest degree
• We have the opposite & the adjacent so we use tan ratio
• Tan A = =
• Tan A =
• Use calculator 2ndF, tan, a/b, 4 ↓ 3 =
• A = 53.1301
• A = 53° Answer 3m adjacent
4m opposite
A°
Oppositeadjacent
43
43
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Solving Right-angle TrianglesSolving Right-angle Triangles• Find the side c (hyp.) to 2 decimal places
• We are given angle & opposite to find hyp.
• sin 40° =
• 0.6428 =
• 0.6428 x c = 15
• c =
• c = 23.34 Answer 23.34 Answer 15m
b
40°
c
15c
15c
150.6428
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Solving Right-angle TrianglesSolving Right-angle Triangles• Given a section through a roof with 2
unequal pitches, calculate
(a) the rise of the roof
(b) the length of rafter c
(c) the span b
30°
b
40°
c
5.42m
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Solving Right-angle TrianglesSolving Right-angle Triangles• This question may seem complicated but
we can simplify it by using our knowledge of right-angle triangles
• Firstly, divide triangle into 2 right-angled triangles (using the rise line)
30°
b
40°
c
5.42m
rise
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Solving Right-angle TrianglesSolving Right-angle Triangles• On the larger triangle we have the angle
30° & the hypotenuse 5.42m so we can use ratio sin 30° to find the opposite (rise)
Sin = (soh)
30°
b
40°
c
5.42m
oppositeoppositehypotenusehypotenuse
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Solving Right-angle TrianglesSolving Right-angle Triangles
• Sin 30°=
• 0.5 =
• 0.5 x 5.42 = rise
• rise = 2.71m Answer (a)2.71m Answer (a)
30°
b
40°
c
5.42m
oppositeoppositehypotenusehypotenuse
rise5.42
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Solving Right-angle TrianglesSolving Right-angle Triangles• So on the smaller triangle we now have the
opposite (2.71m) and the angle 40°• So to find adjacent (part of span) we use Tan
Tan 40° =
0.839 =
0.839 x adj. = 2.71
adj. = 2.71 ÷ 0.839
= 3.23m3.23m
30°b
40°
c
5.42m
2.71adjacent
adj.
OppositeOppositeadjacentadjacent
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Solving Right-angle TrianglesSolving Right-angle Triangles• Using Pythagoras we can find c • a² + b² = c²• 2.71² + 3.23² = c² • 7.3441 + 10.4329 = c²• c² = 17.777• c = √17.777• c = 4.216m Answer (b) 4.216m Answer (b)
30°b
40°
c5.42m
2.71m
3.23m
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Solving Right-angle TrianglesSolving Right-angle Triangles• Using Pythagoras we can find the rest of the span
i.e. the base of the larger triangle• a² + b² = c²
• 2.71² + b² = 5.42²
• 7.344 + b² = 29.376
• b² = 29.376 – 7.344
• b² = 22.032
• b = √22.032
• b = 4.694m4.694m 30° b 40°
c5.42m
2.71m
3.23m
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Solving Right-angle TrianglesSolving Right-angle TrianglesSo 4.694 + 3.23 = span
Span = 7.924m Answer (c)7.924m Answer (c)
30° b 40°
c5.42m
2.71m
3.23m
