Explain Ratios and Find a Length

Check an unfamiliar orientation and explain why a tangent can exceed one.

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Watch the lesson | More practice: opposite over hypotenuse

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Name the selected acute angle before choosing a ratio. Sine and cosine compare a leg with the longer hypotenuse, so they lie between 0 and 1 for acute right-triangle angles. Tangent compares the two legs and can exceed 1. If sine = opposite / hypotenuse is given, multiply the hypotenuse by that ratio to find the opposite length. Keep the stated units.

Worked example

For an acute angle with opposite 21, adjacent 20 and hypotenuse 29, tangent is 21/20, which is greater than 1. For a different model, sine 3/5 and hypotenuse 20 units give opposite = (3/5) x 20 = 12 units.

Right triangle XYZ has a right angle Y, XY 40 units, YZ 9 units and XZ 41 units. The marked acute angle is Z. — Use the chosen angle Z. For the last question, use the new similar triangle and its new hypotenuse.

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 For angle Z in XYZ, what is sin Z? 9/41 41/40 40/41

Question 2 For the same angle Z, what is cos Z? 40/41 9/41 9/40

Question 3 For the same angle Z, what is tan Z? 40/9 40/41 9/40

Question 4 Why is the tangent in the previous question greater than 1? Every ratio involving a triangle must exceed 1 The hypotenuse is shorter than both legs The opposite leg is longer than the adjacent leg

Question 5 A new similar triangle has the same sine 40/41 at its corresponding acute angle and a hypotenuse of 82 cm. How long is its opposite side? 40 cm 80 cm 18 cm

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