5.2e: Exercises - Right Angle Trigonometry (2024)

Table of Contents

A: Given three sides of a right triangle,find all six trigonometric ratios B: Given two sides of right triangle, find all trigonometric ratios of the acute angles C: Given one trigonometric ratio of an acute angle, find all the others D: Cofunctions E: Given one side and an acute angle of a right triangle, find the other sides and angles F: Right Triangle Applications G: Find an angle given 2 sides of a right triangle H: Applications Finding an Angle I: Calculator Practice FAQs References

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A: Given three sides of a right triangle,find all six trigonometric ratios

Exercise \(\PageIndex{A}\)

\( \bigstar \) Given right triangle \(ABC\)where the right angle is angle \(C\) in each figure below,
(a) Label the remaining sides and angles
(b) Designate the hypotenuse, adjacent side or opposite side to angle \(A\).
Determine the trigonometric ratios for (c) \(\sin A\), (d) \(\cosA\), (e) \(\tanA\), (f) \(\secA\), (g) \(\cscA\), (h) \(\cotA\).
Give simplified exact answers - reduce fractions, rationalize all denominators!

B: Given two sides of right triangle, find all trigonometric ratios of the acute angles

Exercise \(\PageIndex{B}\)

\( \bigstar \) Find the exact values of all six trigonometric functions of angles \(A\) and \(B \) in the right triangle \(\triangle\,ABC \) illustrated below. Simplify!

5.\(b = 12 \), \(c = 13\)

6.\(b = 15 \), \(c = 17\)

7.\(a = 7 \), \(c = 25\)

8.\(a = 20 \), \(c = 29\)

9. \(a = 9 \), \(b = 40 \)

10. \(a = 1 \), \(b = 2 \)

11. \(a = 1 \), \(b = 3\)

12. \(a = 2 \), \(b = 5\)

13. \(a = 5 \), \(c = 6\)

14. \(a = 2 \), \(c = \sqrt{6} \)

15. \(b = 7 \), \(c = 8\)

16. \(b = 3 \), \(c = \sqrt{15} \)

\( \bigstar \) In each of the triangles below, find \(\sin \left(A\right), \; \cos \left(A\right),\; \tan \left(A\right),\; \sec \left(A\right),\; \csc \left(A\right),\; \cot \left(A\right)\).

17. 18.

Answers to odd exercises.

For all triangles #5-16, the trigonometric ratios can be determined using the following formulas:
\( \sin A = \cos B = \frac{a}{c},\) \( \cosA = \sinB = \frac{b}{c},\)\( \tan A = \cot B = \frac{a}{b},\)
\( \cscA = \secB = \frac{c}{a},\) \( \secA = \cscB = \frac{c}{b},\)\( \cotA = \tanB = \frac{b}{a},\)
The missing side needed to calculate these ratios is given below:
5. \(a = 5\) 7. \(b = 24\) 9. \(c = 41 \) 11. \( c = \sqrt{10} \) 13. \(b = \sqrt{11} \) 15. a = \( \sqrt{15} \)

17.\(\sin \left(A\right) = \frac{5\sqrt{41}}{41}, \; \cos \left(A\right) = \frac{4\sqrt{41}}{41},\; \tan \left(A\right) = \frac{5}{4},\; \sec \left(A\right) = \frac{\sqrt{41}}{4},\; \csc \left(A\right)= \frac{\sqrt{41}}{5},\; \cot \left(A\right) = \frac{4}{5}\)

C: Given one trigonometric ratio of an acute angle, find all the others

Exercise \(\PageIndex{C}\)

\( \bigstar \) Find the exact values of the other five trigonometric functions of the acute angle \(A\) given the indicated value of one of the functions. Simplify!

21. \( \sin A= \dfrac{1}{2} \)

22. \( \sin A= \dfrac{5}{12} \)

23.\(\cos A= \dfrac{1}{5}\)

24. \(\cos A= \dfrac{2}{3}\)

25. \(\tan A= \dfrac{3}{4}\)

26. \(\tan A= \dfrac{9}{5}\)

27. \(\csc A= \dfrac{7}{3}\)

28. \(\csc A=\dfrac{\sqrt{3}}{1}\)

29. \(\sec A= 3\)

30. \(\sec A= \dfrac{5\sqrt{6}}{3}\)

31. \( \cot A=10\)

32. \(\cot A= 4\)

Answers to odd exercises.: 21. \( \sin A= \frac{1}{2}\), \(\cos A= \frac{\sqrt{3}}{2} \), \( \tanA= \frac{\sqrt{3}}{3}\), \(\csc A= 2\), \( \sec A= \frac{2\sqrt{3}}{3}\), \( \cot A= \sqrt{3}\)
23.\( \sin A= \frac{2\sqrt{6}}{5} \), \( \cos A= \frac{1}{5} \), \( \tan A= 2\sqrt{6} \), \( \csc A= \frac{5\sqrt{6}}{12} \), \( \sec A= 5 \), \( \cot A= \frac{\sqrt{6}}{12}\)
25.\( \sin A= \frac{3}{5} \), \( \cos A= \frac{4}{5} \), \( \tan A= \frac{3}{4} \), \( \csc A= \frac{5}{3} \), \( \sec A= \frac{5}{4} \), \( \cot A= \frac{4}{3} \)
27.\( \sin A= \frac{3}{7} \), \( \cos A= \frac{2\sqrt{10}}{7} \), \( \tan A= \frac{3\sqrt{10}}{20} \), \( \csc A= \frac{7}{3} \), \( \sec A= \frac{7\sqrt{10}}{20} \), \( \cot A= \frac{2\sqrt{10}}{3} \)
29.\( \sin A= \frac{2\sqrt{2}}{3} \), \( \cos A= \frac{1}{3} \), \( \tan A= 2\sqrt{2} \), \( \csc A= \frac{3\sqrt{2}}{4} \), \( \sec A= 3\), \( \cot\theta = \frac{\sqrt{2}}{4} \)
31.\( \sin A= \frac{\sqrt{101}}{101} \), \( \cos A= \frac{10\sqrt{101}}{101} \), \( \tan A= \frac{1}{10} \), \( \csc A=\sqrt{101} \), \( \sec A= \frac{\sqrt{101}}{10} \), \( \cot A= 10\)

D: Cofunctions

Exercise \(\PageIndex{D}\)

Use cofunctions of complementary angles to write an equivalent expression.

36. \( \cos (34°)= \sin (\_\_°)\)

37. \( \cos (\dfrac{π}{3})= \sin (\_\_\_) \)

38. \( \csc (21°) = \sec (\_\_\_°)\)

39. \( \tan (\dfrac{π}{4})= \cot (\_\_)\)

Answers to odd exercises.: 37. \(\dfrac{π}{6}\) 39. \(\dfrac{π}{4}\)

E: Given one side and an acute angle of a right triangle, find the other sides and angles

Exercise \(\PageIndex{E}\)

\( \bigstar \) In each of the following triangles, solve for \(x\) to the nearest tenth.

41.

42.

43.

44.

45.

46.

47.

48.

49.

50.

51.

52.

Answers to odd exercises.: 41. \(6.4\) 43. \(7.7\) 45. \(11.9\) 47. \(8.4\) 49. \(44.8\) 51. \(7.8\)

\( \bigstar \) In each of the following triangles, solve for \(x\) to the nearest tenth.

53.

54.

55.

56.

57.

58.

59.

60,

Answers to odd exercises.: 53. \(20.5\) 55. \(14.5\) 57. \(7.3\) 59. \(4.8\)

\( \bigstar \) In each of the following triangles, solve for the unknown sides and angles. Give answers to 4 decimal digits.

61. 62. 63.

64. 65. 66.

67. 68.

69. \(a=5,\) angle opposite side \(a\) is\( ∡ A=60^∘\)

70. Hypotenuse \(c=12,\) and one acute angle is \( ∡ A=45^∘\)

\( \bigstar \)Find \(x\).Give answers to 4 decimal digits.

71. 72.

73. 74.

Answers to odd exercises.: 61. \(b \approx 12.1244\),\( c =14\),\( B = 60^{\circ} \) 63. \(a \approx 5.3171\),\( c \approx 11.3257\),\( A = 28^{\circ} \) 65. \(a \approx 9.0631\),\( b \approx 4.2262\),\( B = 25^{\circ} \)
67. \(a = 15\),\( b = 15\),\( A = 45^{\circ} \) 69. \(b \approx 2.8868\),\( c \approx 5.7735\),\( B = 30^{\circ} \) 71. \(188.3159\) 73. \(200.6737\)

F: Right Triangle Applications

Exercise \(\PageIndex{F}\)

Solve. Round answers to the nearest \(10^{th}\) unless otherwise specified.

81. A \(23\)-ft ladder leans against a building so that the angle between the ground and the ladder is \(80°\). How high does the ladder reach up the side of the building?

82. A \(33\)-ft ladder leans against a building so that the angle between the ground and the ladder is \(80°\). How high does the ladder reach up the side of the building?

83. A 20 foot ladder is leaning against a wall, It makes an angle of \(70^{\circ}\) with the ground. How high is the top of the ladder from the ground (nearest tenth of a foot)?

84. A 275 foot guy wire is attached to the top of a communication tower. If the wire makes an angle of \(53^{\circ}\) with the ground, how tall is the tower?

ANGLE OF ELEVATION

5.2e: Exercises - Right Angle Trigonometry (39)

85. At a point 50 feet from a tree the angle of elevation of the top of the tree is \(43^{\circ}\). Find the height of the tree to the nearest tenth of a foot.

86. At a point 60 feet from a tree the angle of elevation of the top of the tree is \(40^{\circ}\). Find the height of the tree to the nearest tenth of a foot.

87. The angle of elevation tothe top of a mountain from a point 20 miles away from the base of the mountainis \(6^{\circ}\). How high is the mountain to the nearest foot?Onemile is \(5280\) feet.

5.2e: Exercises - Right Angle Trigonometry (40)

89. At a point 100 feet from a tall building the angle of elevation of the top of the building is \(65^{\circ}\). Find the height of the building to the nearest foot.

90. The angle of elevation to the top of a building in New York is found to be \(9\) degrees from the ground at a distance of \(1\) mile from the base of the building. Using this information, find the height of the building in feet. Onemile is \(5280\) feet.

91. The angle of elevation to the top of a building in Seattle is found to be \(2\) degrees from the ground at a distance of \(2\) miles from the base of the building. Using this information, find the height of the building in feet. Onemile is \(5280\) feet.

93. Assuming that a \(370\)-foot tall giant redwood grows vertically, if I walk a certain distance from the tree and measure the angle of elevation to the top of the tree to be \(60°\), how far from the base of the tree am I?

94. The CN Tower in Toronto is 1815 feet tall. Whenthe angle of elevation to the top of the tower is observed to be \(40°\),how far away from the Tower is that location?

ANGLE OF DEPRESSION

5.2e: Exercises - Right Angle Trigonometry (41)

95. From the top of a building 125 feet tall, the angle of depression of an intersectionis \(34^{\circ} .\) How far from the base of the building is the intersection?
Notice that the angle of elevation will be equal to the corresponding angle of depression.

96. A helicopter that is 700 feet in the air measures the angle of depression to a landing pad as \(24^{\circ} .\) How far is the landing pad from the point directly beneath the helicopter's current position?

5.2e: Exercises - Right Angle Trigonometry (43)

97. From the top of a 100 foot lighthouse the angle of depression of a boat is \(15^{\circ}\). How far is the boat from the bottom of the lighthouse (nearest foot)?

98. From the top of a lighthouse 180 feet above sea level, the angle of depression to a ship in the ocean is \(28^{\circ} .\) How far is the ship from the base of the lighthouse?

5.2e: Exercises - Right Angle Trigonometry (44)

99. From an airplane 5000 feet above the ground the angle of depression of an airport is \(5^{\circ}\). How far away is the airport to the nearest hundred feet?

100. From a helicopter 1000 feet above the ground the angle of depression of a heliport is \(10^{\circ}\). How far away is the heliport to the nearest foot?

TWO ANGLES of DEPRESSIONand/or ELEVATION

101. A radio tower is located \(325\) feet from a building. From a window in the building, a person determines that the angle of elevation to the top of the tower is \(43°\), and that the angle of depression to the bottom of the tower is \(31°\). How tall is the tower?

102. A radio tower is located \(400\) feet from a building. From a window in the building, a person determines that the angle of elevation to the top of the tower is \(36°\), and that the angle of depression to the bottom of the tower is \(23°\). How tall is the tower?

103. A \(400\)-foot tall monument is located in the distance. From a window in a building, a person determines that the angle of elevation to the top of the monument is \(18°\), and that the angle of depression to the bottom of the monument is \(3°\). How far is the person from the monument?

104. A \(200\)-foot tall monument is located in the distance. From a window in a building, a person determines that the angle of elevation to the top of the monument is \(15°\), and that the angle of depression to the bottom of the tower is \(2°\). How far is the person from the monument?

105. There is lightning rod on the top of a building. From a location \(500\) feet from the base of the building, the angle of elevation to the top of the building is measured to be \(36°\). From the same location, the angle of elevation to the top of the lightning rod is measured to be \(38°\). Assuming the lightning rod is situated at the edge of the building facing the observer, find the height of the lightning rod.

106. There is an antenna on the top of a building. From a location \(300\) feet from the base of the building, the angle of elevation to the top of the building is measured to be \(40°\). From the same location, the angle of elevation to the top of the antenna is measured to be \(43°\). Find the height of the antenna.

Answers to odd exercises.: 81. \(22.7\) ft 83. \(18.8\) ft 85. \(46.6\) feet. 87. \(11,099\) feet. 89. \(214\)feet. 91. \(368.8\) ft 93. \(213.6\) ft
95. \(d \approx 185.32\) ft 97. \(373.2\) ft 99. \(57,300\) feet 101. \(498.3\) ft 103. \(1060.1\) ft 105. \(27.4\) ft

G: Find an angle given 2 sides of a right triangle

Exercise \(\PageIndex{G}\)

\( \bigstar \) Find the angle \(x\) to the nearest tenth of a degree:

111.

112.

113.

114.

115.

116.

117.

118.

Answers to odd exercises.: 111. \(41.8^{\circ}\) 113. \(36.9^{\circ}\) 115. \(56.3^{\circ}\) 117. \(48.2^{\circ}\)

H: Applications Finding an Angle

Exercise \(\PageIndex{H}\)

121. If a 20 foot telephone pole casts a shadow of 43 feet , what is the angle of elevation of the sun to the nearest tenth?

122. An 88 foot tree casts a shadow that is 135 feet long. What is the angle of elevation of the sun to the nearest tenth?

123. A road rises 30 feet in a horizontal distance of 300 feet. Find the angle the road makes with the horizontal to the nearest tenth of a degree.

124. A road rises 10 feet in a horizontal distance of 400 feet. Find the angle the road makes with the horizontal to the nearest tenth of a degree.

Answers to odd exercises.: 121. \(24.9^{\circ} \) 123. \(5.7^{\circ}\)

I: Calculator Practice

Exercise \(\PageIndex{I}\)

\( \bigstar \) Use a calculator to find the value to four decimal digits. Be sure your calculator is in the appropriate mode!

131. \(\sin 10^{\circ}\) 132. \(\sin 30^{\circ}\) 133. \(\cos 80^{\circ}\) 134. \(\cos 60^{\circ}\) 135. \(\tan 45^{\circ}\)

136. \(\tan 60^{\circ}\) 137. \(\sin 18^{\circ}\) 138. \(\cos 72^{\circ}\) 139. \(\tan 50^{\circ}\) 140. \(\tan 80^{\circ}\)

141. \(\sin \left( \dfrac{\pi}{12} \right) \) 142. \(\sin \left( \dfrac{5\pi}{12} \right) \) 143. \(\cos\left( \dfrac{3\pi}{8} \right) \) 144. \(\cos\left( \dfrac{3\pi}{8} \right) \)
145. \(\tan \left( \dfrac{\pi}{3} \right) \) 146. \(\tan \left( 0.8\right) \) 147. \(\sin \left( 1.5\right) \) 148. \(\cos\left( 1.0\right) \)

Answers to odd exercises.: 131. \(0.1736\) 133. \(0.1736\) 135. \(1\) 137. \(0.3090\) 139. \(1.1918\)
141. \(0.2588\) 143. \(0.3827\) 145. \(1.7321\) 147. \(0.9975\)

\( \bigstar \) Use a calculator to find the value (a) in degrees and (b) in radians to four decimal digits.

151. \(\sin^{-1} (.1)\) 152. \(\sin^{-1} (.1)\) 153. \(\cos^{-1} (.3)\) 154. \(\cos^{-1} (.9)\)

155. \(\tan^{-1} (.5)\) 156. \(\tan^{-1} (4)\) 157. \(\sin^{-1} (3)\) 158. \(\cos^{-1} (2)\)

Answers to odd exercises.: 151. (a) \(6.7392^{\circ}\) (b)\(.1002\) (radians) 153. (a) \(72.5425^{\circ}\) (b)\(1.2661\) (radians)
155. (a) \(26.5651^{\circ}\) (b) \(1.3258\) (radians) 157. Undefined ("Domain Error")

5.2e: Exercises - Right Angle Trigonometry (2024)

FAQs

What is the formula for right angle trigonometry? ›

Solving right triangles

Pythagorean theorem: a² + b² = c². Sines: sin A = a/c, sin B = b/c. Cosines: cos A = b/c, cos B = a/c. Tangents: tan A = a/b, tan B = b/a.

Can you use trig functions on non-right triangles? ›

Explanation: Although most often trigonometric functions are used with right triangles there are some situations when they can be used for any type of triangle. Examples: If you have two sides given and an angle between them you can use the trigonometric functions the Law of Cosines to calculate the third side.

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What are the sides of a right angle triangle? ›

A right triangle has one angle equal to 90° and the other two angles are acute angles. The side opposite to the right angle is the largest side and is referred to as the hypotenuse. The other two sides of a right triangle are called the legs and are specifically referred to as the 'base and 'height' of the triangle.

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What is opposite over hypotenuse? ›

Let us start with some definitions. We will call the ratio of the opposite side of a right triangle to the hypotenuse the sine and give it the symbol sin. sin = o / h. The ratio of the adjacent side of a right triangle to the hypotenuse is called the cosine and given the symbol cos. cos = a / h.

What are the 45 formulas of trigonometry? ›

List of Trigonometry Formulas

sin²θ + cos²θ = 1.
tan²θ + 1 = sec²θ
cot²θ + 1 = cosec²θ
sin 2θ = 2 sin θ cos θ
cos 2θ = cos²θ – sin²θ
tan 2θ = 2 tan θ / (1 – tan²θ)
cot 2θ = (cot²θ – 1) / 2 cot θ

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Is Soh Cah Toa only for right triangles? ›

The SOHCAHTOA method is used to find a side or angle in a right-angled triangle. The longest side of the right-angled triangle is called the hypotenuse. The side opposite the angle we are using is labelled opposite, and the remaining side next to the angle is labelled adjacent.

How many formulas are there in trigonometry? ›

The six trigonometric functions are sine, cosine, secant, cosecant, tangent and cotangent. By using a right-angled triangle as a reference, the trigonometric functions and identities are derived: sin θ = Opposite Side/Hypotenuse. cos θ = Adjacent Side/Hypotenuse.

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Is trigonometry for right angles only? ›

These formulas work for any triangle whether acute, obtuse, or right. We'll use the standard notation where the three vertices of the triangle are denoted with the uppercase letters A, B, and C, while the three sides opposite them are respectively denoted with lowercase letters a, b, and c.

Learn More Now ›

What are the 3 parts of a right-angled triangle? ›

In a right triangle, the hypotenuse is the longest side, an "opposite" side is the one across from a given angle, and an "adjacent" side is next to a given angle.

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Which is the longest side in a right-angled triangle? ›

In a right triangle, hypotenuse is the longest side.

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What is the formula for a right angle triangle? ›

Pythagoras Theorem - Formula: (Hypotenuse)² = (Perpendicular)² + (Base) Area of a right triangle formula: Area = 1/2 × Base × Height. Perimeter of a right triangle formula = Sum of lengths of 3 sides.

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What is the tan rule in trigonometry? ›

The tangent rule states that the ratio of difference and sum of any two sides of a triangle is equal to the ratio of the tangent of half the difference and tangent of sum of the angles opposite to these sides.

Is tan short for tangent? ›

Finally, the ratio of the opposite side to the adjacent side is called the tangent and given the symbol tan.

What is the formula for a 90 degree triangle? ›

These formulas are given as: Pythagoras Theorem - Formula: (Hypotenuse)² = (Perpendicular)² + (Base) Area of a right triangle formula: Area = 1/2 × Base × Height. Perimeter of a right triangle formula = Sum of lengths of 3 sides.

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How do I find a right angle? ›

When two straight lines intersect each other at 90˚ or are perpendicular to each other at the intersection, they form the right angle.

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What is the formula for right angle properties? ›

Pythagoras Theorem (Formula): (Hypotenuse)² = (Perpendicular)² + (Base) Area of Right Triangle Formula: Area = 1/2 × Base × Height. Perimeter of Right Triangle Formula: Sum of lengths of 3 sides.

Read On ›

What is the formula for the sine angle? ›

Sine θ = Opposite side/Hypotenuse = BC/AC. Cos θ = Adjacent side/Hypotenuse = AB/AC. Tan θ = Opposite side/Adjacent side = BC/AB.

Tell Me More ›