08.1 - Sine, Cosine, and Tangent

Find the sine, cosine, and tangent ratios for angle A and for angle B in the following triangle. (Hint: Use the Pythagorean theorem to find the missing side in the triangle first.) Triangle: Right triangle with sides AB = ?, BC = 8, AC = 15.

• sin A = 8/17

  sin B = 15/17

  cos A = 15/17

  cos B = 8/17

  tan A = 8/15

  tan B = 15/8

• sin A = 15/17

  sin B = 8/17

  cos A = 8/17

  cos B = 15/17

  tan A = 15/8

  tan B = 8/15

• sin A = 8/15

  sin B = 15/8

  cos A = 15/8

  cos B = 8/15

  tan A = 8/17

  tan B = 15/17

• sin A = 15/8

  sin B = 8/15

  cos A = 8/15

  cos B = 15/8

  tan A = 17/8

  tan B = 17/15

Answer explanation

Using the Pythagorean theorem, the missing side AB is 17. For angle A, sin A = opposite/hypotenuse = 8/17, cos A = adjacent/hypotenuse = 15/17, tan A = opposite/adjacent = 8/15. For angle B, sin B = 15/17, cos B = 8/17, tan B = 15/8.

Tags

CCSS.HSG.SRT.C.6

Find the sine, cosine, and tangent ratios for angle A and for angle B in the following triangle. (Hint: Use the Pythagorean theorem to find the missing side in the triangle first.) Triangle: Right triangle with sides AB = ?, BC = ?, AC = 24, BC = 7.

• sin A = 7/25

  sin B = 24/25

  cos A = 24/25

  cos B = 7/25

  tan A = 7/24

  tan B = 24/7

• sin A = 24/25

  sin B = 7/25

  cos A = 7/25

  cos B = 24/25

  tan A = 24/7

  tan B = 7/24

• sin A = 24/25

  sin B = 7/25

  cos A = 24/25

  cos B = 7/25

  tan A = 7/24

  tan B = 24/7

• sin A = 7/24

  sin B = 24/25

  cos A = 24/25

  cos B = 7/25

  tan A = 7/25

  tan B = 24/7

Answer explanation

Using the Pythagorean theorem, the missing side AB is 25. For angle A, sin A = opposite/hypotenuse = 7/25, cos A = adjacent/hypotenuse = 24/25, and tan A = opposite/adjacent = 7/24. For angle B, sin B = 24/25, cos B = 7/25, tan B = 24/7.

Tags

CCSS.HSG.SRT.C.6

Find the sine, cosine, and tangent ratios for angle A and for angle B in the following triangle. (Hint: Use the Pythagorean theorem to find the missing side in the triangle first.) Triangle: Right triangle with sides AB = 15, BC = ?, AC = 12.

• sin A = 3/5

  sin B = 4/5

  cos A = 4/5

  cos B = 3/5

  tan A = 3/4

  tan B = 4/3

• sin A = 4/5

  sin B = 3/5

  cos A = 3/5

  cos B = 4/5

  tan A = 4/3

  tan B = 3/4

• sin A = 12/15

  sin B = 15/12

  cos A = 15/12

  cos B = 12/15

  tan A = 12/9

  tan B = 15/9

• sin A = 15/12

  sin B = 12/15

  cos A = 12/15

  cos B = 15/12

  tan A = 9/12

  tan B = 9/15

Answer explanation

Using the Pythagorean theorem, find BC = 9. For angle A, sin A = opposite/hypotenuse = 9/15 = 3/5, cos A = adjacent/hypotenuse = 12/15 = 4/5, tan A = opposite/adjacent = 9/12 = 3/4. For angle B, sin B = 12/15 = 4/5, cos B = 9/15 = 3/5, tan B = 12/9 = 4/3.

Tags

CCSS.HSG.SRT.C.6

Find the sine, cosine, and tangent ratios for angle A and for angle B in the following triangle. (Hint: Use the Pythagorean theorem to find the missing side in the triangle first.) Triangle: Right triangle with sides AB = ?, BC = 13, AC = 12.

• sin A = 12/17

  sin B = 13/17

  cos A = 13/17

  cos B = 12/17

  tan A = 12/13

  tan B = 13/12

• sin A = 13/17

  sin B = 12/17

  cos A = 12/17

  cos B = 13/17

  tan A = 13/12

  tan B = 12/13

• sin A = 5/13

  sin B = 12/13

  cos A = 12/13

  cos B = 5/13

  tan A = 5/12

  tan B = 12/5

• sin A = 12/13

  sin B = 5/13

  cos A = 5/13

  cos B = 12/13

  tan A = 12/5

  tan B = 5/12

Answer explanation

Using the Pythagorean theorem, the missing side AB is 5. For angle A, sin A = opposite/hypotenuse = 12/13, cos A = adjacent/hypotenuse = 5/13, tan A = opposite/adjacent = 12/5. For angle B, sin B = 5/13, cos B = 12/13, tan B = 5/12.

Tags

CCSS.HSG.SRT.C.6

Use right-triangle trigonometry to find the value for x, without calculator. Once you have solved for x without calculator, use a scientific calculator to find the value of x to the nearest thousandth (i.e. 3 decimal place accuracy). Triangle: Right triangle with sides AB = 18, angle BAC = 25°, AC = x.

• 5.432

• 10.256

• 16.314

• 12.789

Answer explanation

Using the sine function, sin(25°) = opposite/hypotenuse = AB/x. Thus, x = AB/sin(25°) = 18/sin(25°) ≈ 16.314. The correct answer is 16.314.

Tags

CCSS.HSG.SRT.C.8

Use right-triangle trigonometry to find the value for x, without calculator. Once you have solved for x without calculator, use a scientific calculator to find the value of x to the nearest thousandth (i.e. 3 decimal place accuracy). Triangle: Right triangle with angle BAC = 20°, AC = x, BC = 8.

• x = 8 / sin(20°) ≈ 23.399

• x = 8 * sin(20°) ≈ 2.736

• x = 8 / cos(20°) ≈ 8.514

• x = 8 * cos(20°) ≈ 7.519

Answer explanation

In triangle ABC, angle BAC = 20° and side BC = 8. To find AC (x), use the sine function: sin(20°) = opposite/hypotenuse = 8/x. Rearranging gives x = 8/sin(20°). Thus, the correct choice is x = 8 / sin(20°) ≈ 23.399.

Tags

CCSS.HSG.SRT.C.8

Use right-triangle trigonometry to find the value for x, without calculator. Once you have solved for x without calculator, use a scientific calculator to find the value of x to the nearest thousandth (i.e. 3 decimal place accuracy). Triangle: Right triangle with angle BAC = x°, AC = 8, BC = 13.

• x = arccos(8/13) ≈ 50.317°

• x = arcsin(8/13) ≈ 38.682°

• x = arccos(13/8) ≈ 58.482°

• x = arctan(8/13) ≈ 31.744°

Answer explanation

In triangle ABC, with AC = 8 and BC = 13, we use arctan to find angle x. Since tan(x) = opposite/adjacent = AC/BC = 8/13, we have x = arctan(8/13) ≈ 31.744°.

Tags

CCSS.HSG.SRT.C.8

Use right-triangle trigonometry to find the value for x, without calculator. Once you have solved for x without calculator, use a scientific calculator to find the value of x to the nearest thousandth (i.e. 3 decimal place accuracy). Triangle: Right triangle with angle BCA = x°, AB = 15, BC = 13.

• x = arcsin(13/15) ≈ 60.069°

• x = arccos(13/15) ≈ 29.932°

• x = arctan(15/13) ≈ 49.398°

• x = arcsin(15/13) (undefined)

Answer explanation

In triangle BCA, use the sine function: sin(x) = opposite/hypotenuse = 13/15. Thus, x = arcsin(13/15). This gives approximately 60.069°, confirming the correct choice.

Tags

CCSS.HSG.SRT.C.8

Use right-triangle trigonometry to find the value for x, without calculator. Once you have solved for x without calculator, use a scientific calculator to find the value of x to the nearest thousandth (i.e. 3 decimal place accuracy). Triangle: Right triangle with angle BAC = 58°, AC = x, BC = 9√7.

• x = 9√7 / sin(58°) ≈ 10.682

• x = 9√7 * sin(58°) ≈ 67.123

• x = 9√7 / cos(58°) ≈ 44.935

• x = 9√7 * cos(58°) ≈ 33.789

Answer explanation

In triangle ABC, use cosine for adjacent side AC. Thus, x = BC / cos(58°) = 9√7 / cos(58°). This gives the correct choice x ≈ 44.935.

Tags

CCSS.HSG.SRT.C.8

Use right-triangle trigonometry to find the value for x, without calculator. Once you have solved for x without calculator, use a scientific calculator to find the value of x to the nearest thousandth (i.e. 3 decimal place accuracy). Triangle: Right triangle with angle CBA = x°, AB = 25, BC = 4√23.

• x = arccos(4√23/25) ≈ 40°

• x = arcsin(4√23/25) ≈ 30°

• x = arctan(4√23/25) ≈ 45°

• x = arccos(25/4√23) ≈ 75°

Answer explanation

In triangle ABC, use cosine: cos(x) = adjacent/hypotenuse = BC/AB = (4√23)/25. Thus, x = arccos(4√23/25). This matches the correct choice, approximately 40°.

Tags

CCSS.HSG.SRT.C.8