Functions of a right triangle
In right triangle ABC with the right angle at C, name each side by the angle it faces: side a is opposite angle A, side b is opposite angle B, and the hypotenuse c is opposite the right angle.
Sine and cosecant
Cosine and secant
Tangent and cotangent
Pythagorean theorem
Special angles (memorize these)
| Angle | sin | cos | tan |
|---|---|---|---|
| 30° | |||
| 45° | 1 | ||
| 60° |
In right triangle ABC (C = 90°), and . Find b and all six trigonometric functions of angle A.
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Basic trigonometric identities
Identities are equations that are true for every angle. Use them to simplify expressions and to find one function from another.
Reciprocal
Quotient
Pythagorean
Cofunction
Negative angle
Sum and difference
Double angle
Half angle
Simplify .
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Write everything in sine and cosine:
sin θIf and θ is in Quadrant II, find and .
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In Quadrant II, cosine is negative:
sin 2θ = −24/25, cos 2θ = 7/25Oblique triangles
An oblique triangle has no right angle. Solve it with the Law of Sines or the Law of Cosines, depending on what is given.
Law of Sines
R = radius of the circumscribed circle
Law of Cosines
| Given | Use first |
|---|---|
| Two angles and a side (ASA, AAS) | Law of Sines |
| Two sides and the included angle (SAS) | Law of Cosines |
| Three sides (SSS) | Law of Cosines |
| Two sides and an angle opposite one of them (SSA) | Law of Sines — check the ambiguous case |
A triangle has sides , and . Find its three angles.
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In triangle ABC, , and the included angle . Solve the triangle.
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In triangle ABC, , and the side between them is . Solve the triangle.
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In triangle ABC, , and . Solve the triangle.
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Given , and . How many triangles are possible? Solve them.
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Since (h < a < b), there are two triangles.
Triangle 1:
Triangle 2:
Two trianglesArea of triangles
Pick the formula that matches what the problem gives you.
Base and height
Two sides and the included angle
Three sides (Heron's formula)
One side and all angles
With the inscribed circle (radius r)
With the circumscribed circle (radius R)
Find the area of the triangle with sides 7, 8 and 9 in two ways.
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Heron's formula:
Two sides and the included angle (C = 73.40° from Example 4):
Area = 26.83 square unitsTwo sides of a triangular lot are 40 m and 55 m, and the angle between them is 62°. Find the area of the lot.
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Angles of elevation and depression
Both angles are measured from the horizontal. Looking up at an object gives the angle of elevation. Looking down at an object gives the angle of depression.
From a point 50 m from the foot of a building, the angle of elevation of the top is 38°. How tall is the building?
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From the top of a lighthouse 60 m high, the angle of depression of a boat is 25°. How far is the boat from the foot of the lighthouse?
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The angle of elevation from the boat to the top is also 25°.
128.67 mFrom point A, the angle of elevation of the top of a tower is 30°. After walking 40 m straight toward the tower to point B, the angle of elevation becomes 45°. Find the height of the tower.
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Let x = distance from B to the foot of the tower and h = height.
h = 54.64 mApplication problems
Board problems describe a real situation. Draw a sketch first, label what is given, then choose the right formula.
To find the distance between points A and B on opposite banks of a river, a surveyor measures AC = 120 m along one bank, angle BAC = 70° and angle BCA = 62°. Find AB.
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Two ships leave a port at the same time. One sails at 20 kph on bearing N 30° E; the other sails at 25 kph on bearing S 70° E. How far apart are they after 2 hours?
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Distances after 2 hours: 40 km and 50 km. Measured from north, the directions are 30° and 180° − 70° = 110°, so the angle between the paths is 80°.
58.36 kmA straight road rises at an angle of 6° with the horizontal. How much higher is a car after travelling 800 m along the road?
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A 12 m pole casts a shadow 7 m long. Find the angle of elevation of the sun.
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Quiz: test yourself
Pick an answer to check it right away. Open the solution when you want to see the steps.
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