Choose Sine, Cosine or Tangent

Match the side roles to each ratio before using the lengths.

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Watch the lesson | Next: ratios above one and a missing length

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For an acute angle in a right triangle: sine = opposite / hypotenuse, cosine = adjacent leg / hypotenuse, tangent = opposite / adjacent leg. Choose the angle and label the roles before dividing. All lengths must use the same unit. Scaling every side by the same positive factor preserves these ratios. The diagram is schematic.

Worked example

For an acute angle with opposite 7, adjacent 24 and hypotenuse 25, sine is 7/25, cosine is 24/25 and tangent is 7/24. The angle at the other end of the hypotenuse exchanges the opposite and adjacent leg roles.

Right triangle UVW has a right angle V, UV 35 units, VW 12 units and UW 37 units. Angle U is marked. — Use UVW unless a question describes its scaled copy. The chosen angle is U unless stated otherwise.

Would you believe it?

In a right triangle, the two acute angles add to 90 degrees. Sine of one equals cosine of the other because the same leg is opposite one angle and adjacent to the other.

OpenStax: complementary-angle ratios

Question 1 What is sin U? 35/37 12/37 37/12

Question 2 What is cos U? 35/37 12/35 37/35

Question 3 What is tan U? 35/12 12/37 12/35

Question 4 A similar copy doubles every side. What is the sine of its angle corresponding to U? 24/37, because only the opposite doubles 24/74, which equals 12/37 12/74, because only the hypotenuse doubles

Question 5 Now choose angle W in the original triangle. What is cos W? 35/37 12/35 12/37

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