1. In the right triangle PQR shown below, find the six trigonometric ratios of the angle θ.
2. In the figure shown below, find the six trigonometric ratios of the angle θ.
3. In triangle ABC, right angled at B, 15sinA = 12. Find the other five trigonometric ratios of the angle A.
4. In the figure shown below, find the values of
sinB, secB, cotB, cosC, tanC and cscC
5. Find the length of the missing side (x). Round to the nearest tenth.
6. Find the length of the missing side (x). Round to the nearest tenth.
1. Answer :
In the above right angled triangle, note that for the given angle θ, PQ is the ‘opposite’ side and PR is the ‘adjacent’ side.
Then,
sinθ = opposite side/hypotenuse = PQ/QR = 5/13
cosθ = adjacent side/hypotenuse = PR/QR = 12/13
tanθ = opposite side/adjacent side = PQ/PR = 5/12
cscθ = 1/sinθ = 13/5
secθ = 1/cosθ = 13/12
cotθ = 1/tanθ = 12/5
2. Answer :
In the right angled triangle ABC shown above,
AC = 24
BC = 7
By Pythagorean theorem,
AB^{2} = BC^{2} + CA^{2}
AB^{2} = 7^{2} + 24^{2}
AB^{2} = 49 + 576
AB^{2} = 625
AB^{2} = 25^{2}
AB = 25
Now, we can use the three sides to find the six trigonometric ratios of angle θ.
sinθ = opposite side/hypotenuse = BC/AB = 7/25
cosθ = adjacent side/hypotenuse = AC/AB = 24/25
tanθ = opposite side/adjacent side = BC/AC = 7/24
cscθ = 1/sinθ = 25/7
secθ = 1/cosθ = 25/24
cotθ = 1/tanθ = 24/7
3. Answer :
15sinA = 12
sinA = 12/15
sinA = opposite side/hypotenuse
sinA = 12/15
By Pythagorean theorem,
AC^{2} = AB^{2} + BC^{2}
15^{2} = AB^{2} + 12^{2}
225 = AB^{2} + 144
Subtract 144 from each side.
81 = AB^{2}
9^{2} = AB^{2}
9 = AB
Now, we can use the three sides to find the five trigonometric ratios of angle A and six trigonometric ratios of angle C.
cosA = adjacent side/hypotenuse
= AB/AC
= 9/15
= 3/5
tanA = opposite side/adjacent side
= BC/AB
= 12/9
= 4/3
cscA = 1/sinA
= 15/12
= 5/4
secA = 1/cosA
= 5/3
cotA = 1/tanA
= 3/4
4. Answer :
In the right ΔABD, by Pythagorean Theorem,
AB^{2} = AD^{2} + BD^{2}
13^{2} = AD^{2} + 5^{2}
169 = AD^{2} + 25
Subtract 25 from each side.
144 = AD^{2}
12^{2} = AD^{2}
12 = AD
In the right ΔACD, by Pythagorean Theorem,
AC^{2} = AD^{2} + CD^{2}
AC^{2} = 12^{2} + 16^{2}
AC^{2} = 144 + 256
AC^{2} = 400
AC^{2} = 20^{2}
AC = 20
Then,
sinB = opposite side/hypotenuse = AD/AB = 12/13
secB = hypotenuse/adjacent side = AB/BD = 13/5
cotB = adjacent side/opposite side = BD/AD = 5/12
cosC = adjacent side/hypotenuse
= CD/AC
= 16/20
= 4/5
tanC = opposite side/adjacent side
= AD/CD
= 12/16
= 3/4
cscC = hypotenuse/opposite side
= AC/AD
= 20/12
= 5/3
5. Answer :
In the triangle shown above, the length of the hypotenuse is x and for the angle 51°, the side which has the length 10 is adjacent side.
We know the length of adjacent side and we have to find the length of hypotenuse.
So, we have to use the trigonometric ratio in which we have hypotenuse and adjacent side.
sec51° = hypotenuse/adjacent side
sec51° = x/10
Multiply each side by 10.
10 ⋅ sec51° = x
Use calculator.
15.9 ≈ x
6. Answer :
In the triangle shown above, the length of the hypotenuse is 17 and for the angle 59°, the side which has the length x is opposite side.
We know the length of hypotenuse and we have to find the length of opposite side.
So, we have to use the trigonometric ratio in which we have opposite side and hypotenuse.
Then, we have
sin59° = opposite side/hypotenuse
sin59° = x/17
Multiply each side by 17.
17 ⋅ sin59° = x
Use calculator.
14.6 ≈ x
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