Podcast
Questions and Answers
When solving a triangle given two sides and the included angle, the Law of ______ is typically the best choice.
When solving a triangle given two sides and the included angle, the Law of ______ is typically the best choice.
Cosines
If triangle $ABC$ has sides a = 17, c = 34, and angle $B = 94°$, the length of side b can be found using the Law of ______.
If triangle $ABC$ has sides a = 17, c = 34, and angle $B = 94°$, the length of side b can be found using the Law of ______.
Cosines
For an isosceles triangle with a base of 12 feet and a vertex angle of 65°, the area can be calculated using trigonometric functions to find the height, which, when combined with the base, gives the triangle's ______.
For an isosceles triangle with a base of 12 feet and a vertex angle of 65°, the area can be calculated using trigonometric functions to find the height, which, when combined with the base, gives the triangle's ______.
area
When determining the largest angle in a triangle given all three sides, such as the triangle formed by the locations of Peter, Jamie, and Diane's factories, you would use the Law of ______.
When determining the largest angle in a triangle given all three sides, such as the triangle formed by the locations of Peter, Jamie, and Diane's factories, you would use the Law of ______.
To find the length of a side in a triangle when you know two sides and the included angle, as in triangle $HAT$ where $a = 7.4$, $t = 10.2$, and angle $H = 67°$, it's appropriate to use the law of ______.
To find the length of a side in a triangle when you know two sides and the included angle, as in triangle $HAT$ where $a = 7.4$, $t = 10.2$, and angle $H = 67°$, it's appropriate to use the law of ______.
In an isosceles triangle $QRS$ where $q = s = 5.7$ cm and $\cos R = 0.2908$, finding the length of side $r$ involves applying the Law of ______.
In an isosceles triangle $QRS$ where $q = s = 5.7$ cm and $\cos R = 0.2908$, finding the length of side $r$ involves applying the Law of ______.
To find the length of a side opposite an angle in a triangle given two sides and an included angle, you use the Law of ______.
To find the length of a side opposite an angle in a triangle given two sides and an included angle, you use the Law of ______.
To calculate the ______ of a triangle when two sides and the included angle are known, the formula $A = \frac{1}{2}qr \sin P$ is used, where $q$ and $r$ are the sides and $P$ is the angle.
To calculate the ______ of a triangle when two sides and the included angle are known, the formula $A = \frac{1}{2}qr \sin P$ is used, where $q$ and $r$ are the sides and $P$ is the angle.
To determine the height of a building using angles of elevation from a point on the ground, along with the additional height of a sign on top, trigonometric functions like sine, cosine, or ______ are applied.
To determine the height of a building using angles of elevation from a point on the ground, along with the additional height of a sign on top, trigonometric functions like sine, cosine, or ______ are applied.
In solving for the length of a ski lift given its angle of ascent and the angle of elevation of the mountain face, as well as the horizontal distance from the base, the Law of ______ is applied.
In solving for the length of a ski lift given its angle of ascent and the angle of elevation of the mountain face, as well as the horizontal distance from the base, the Law of ______ is applied.
To find an angle within a trapezoid using the dimensions of the sides a ratio using the ______ function can be used.
To find an angle within a trapezoid using the dimensions of the sides a ratio using the ______ function can be used.
To calculate how high a ladder on a fire truck can reach, given its length and the maximum angle of elevation, the ______ function is used.
To calculate how high a ladder on a fire truck can reach, given its length and the maximum angle of elevation, the ______ function is used.
Determining the height of an office building when sighting the top and bottom from a distance involves using angles of elevation and ______, along with trigonometric functions such as tangent.
Determining the height of an office building when sighting the top and bottom from a distance involves using angles of elevation and ______, along with trigonometric functions such as tangent.
To determine the height of a larger mountain when viewed from a smaller mountain, using angles to the base and top of the larger mountain, trigonometric relationships involving sine, cosine, and ______ are used.
To determine the height of a larger mountain when viewed from a smaller mountain, using angles to the base and top of the larger mountain, trigonometric relationships involving sine, cosine, and ______ are used.
In triangle $ABC$, to solve for side $x$ when given angle $50°$, side 32, and side 40, one must use the Law of ______.
In triangle $ABC$, to solve for side $x$ when given angle $50°$, side 32, and side 40, one must use the Law of ______.
Given a triangle with information such as two sides and an angle, determining which law--Law of Sines or Law of Cosines--to utilize depends on what you need to ______.
Given a triangle with information such as two sides and an angle, determining which law--Law of Sines or Law of Cosines--to utilize depends on what you need to ______.
The Law of ______ is useful and should be applied when solving for missing angles in a triangle.
The Law of ______ is useful and should be applied when solving for missing angles in a triangle.
The Law of ______ relates the lengths of the sides of a triangle to the cosine of one of its angles
The Law of ______ relates the lengths of the sides of a triangle to the cosine of one of its angles
When solving a triangle, remember to consider the unit of the answer to ensure that it is ______.
When solving a triangle, remember to consider the unit of the answer to ensure that it is ______.
In the Law of Sines and Law of Cosines formulas, knowing both sides and angles leads to a solvable ______.
In the Law of Sines and Law of Cosines formulas, knowing both sides and angles leads to a solvable ______.
Flashcards
Law of Sines
Law of Sines
States that in any triangle, the ratio of the length of a side to the sine of its opposite angle is constant.
Law of Cosines
Law of Cosines
Relates the lengths of the sides of a triangle to the cosine of one of its angles.
Isosceles Triangle Properties
Isosceles Triangle Properties
In an isosceles triangle, two sides have equal length, and the angles opposite those sides are also equal.
Area of a Triangle with Sine
Area of a Triangle with Sine
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Angle Sum of a Triangle
Angle Sum of a Triangle
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Law of Cosines for Angles
Law of Cosines for Angles
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Angle of Elevation
Angle of Elevation
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Angle of Depression
Angle of Depression
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Solving with Trig Ratios
Solving with Trig Ratios
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Study Notes
- To solve for a missing side or angle in a triangle, determine whether to use the Law of Sines or Law of Cosines.
- The Law of Sines is used to solve for x in the triangle, by setting up the proportion 40 / sin(50) = 32 / sin(x).
- Solving the proportion gives x = sin^-1((32*sin(50))/40) ≈ 37.8°.
- The Law of Cosines is used to find the length of side b in triangle ABC, given a = 17, c = 34, and angle B = 94°.
- Using the Law of Cosines formula b^2 = a^2 + c^2 - 2ac * cos(B) gives b^2 = 17^2 + 34^2 - 2(17)(34)cos(94).
- Solving for b gives b ≈ 39.06.
- The area of an isosceles triangle with a base of 12 feet and a vertex angle of 65° is found using the formula A = 1/2 * a * b * sin(C).
- Since the base angles are equal, each is (180 - 65) / 2 = 57.5°.
- The length of the equal sides is found using the Law of Sines: 12 / sin(65) = b / sin(57.5), so b = (12 * sin(57.5)) / sin(65). The area is A = 0.5 * 12 * ((12*sin(57.5))/sin(65)) * sin(57.5) ≈ 57 ft².
- Peter and Jamie have computer factories 132 miles apart; Diane's factory is 72 miles from Peter and 84 miles from Jamie.
- The largest angle is found using the Law of Cosines: 132^2 = 84^2 + 72^2 - 2 * 84 * 72 * cos(D).
- Solving for angle D gives D = cos^-1((5184) / (-12096)) ≈ 115.4°.
- In triangle HAT, given a = 7.4, t = 10.2, and angle H = 67°, to find the length of side h, use the Law of Cosines.
- h^2 = a^2 + t^2 - 2 * a * t * cos(H) which gives h = √(7.4^2 + 10.2^2 - 2 * 7.4 * 10.2 * cos(67)) ≈ 10.0.
- In isosceles triangle QRS, q = s = 5.7 cm and cos(R) = 0.2908, the length of side r is found using the Law of Cosines.
- r^2 = q^2 + s^2 - 2 * q * s * cos(R) which gives r = √(5.7^2 + 5.7^2 - 2 * 5.7 * 5.7 * 0.2908) ≈ 6.79 cm.
- A race course forms a triangle PQR; participants run 1.4 miles from P to Q, then to R, and 2.6 miles back to P; angle QPR is 38.5 degrees.
- The total distance equals p + q + r.
- Find p using the Law of Cosines: p^2 = r^2 + q^2 - 2 * r * q * cos(P), so p = √(1.4^2 + 2.6^2 - 2 * 1.4 * 2.6 * cos(38.5)) ≈ 1.9 miles.
- Total distance is approximately 1.9 + 1.4 + 2.6 = 5.9 miles.
- The area of triangle PQR is found using the formula Area = 0.5 * q * r * sin(P) = 0.5 * 1.4 * 2.6 * sin(38.5) ≈ 1.1 square miles.
- A sign 46 feet high tops an office building; the angle of elevation to the top of the sign is 40°, and to the bottom is 32°.
- Use right triangle trigonometry to find the building's height.
- sin y/x, or 46/Sin50 = x/Sin8.
- This calculates the height of the building as follows: y = (46*sin(32))/sin(8) which is approximately 134 feet.
- A ski lift starts 0.75 miles from a mountain base at a 50° angle, ascending at 20°.
- Find the ski lift length.
- Set up the equation using the law of sines 0.75/sin30 = X/Sin130.
- x = 0.75*sin(130)/sin(30) is about 1.15 miles.
- Finding the measure of angle x in the trapezoid requires using the sine function.
- sin(x) = opposite / hypotenuse = 8/12.
- Therefore, x = sin^-1(8/12) ≈ 41.8 degrees.
- A fire truck ladder is based 8 feet above ground, and its maximum length is 100 feet; the greatest elevation angle is 70°.
- To find the highest reach, use sin(70) = (y-8)/100. Then y = 100*sin(70) + 8 ≈ 102 feet.
- In an apartment building, the top of an office building 500 feet away is at an elevation angle of 23°, and the base at a depression angle of 50°. The building’s height can be determined by using the equation x = y + z, where tan(23) = y/500 and tan(50) = z/500.
- Solving for height x = 500tan(23) + 500tan(50) ≈ 808.1 feet.
- From a small mountain, at a larger one 2 miles away, the angle of depression to the base is 24°, and the elevation angle to the top is 14°.
- The larger mountain's height calculation uses sin(14) = x/2, tan(24) = y/(2cos(14)), and z ≈ x + y ≈ 1.35 miles.
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