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Right Triangle Trigonometry

θ. Right Triangle Trigonometry. Trigonometry is based upon ratios of the sides of right triangles. The six trigonometric functions of a right triangle, with an acute angle  , are defined by ratios of two sides of the triangle. hyp. opp. adj. The sides of the right triangle are:.

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Right Triangle Trigonometry

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  1. θ Right Triangle Trigonometry Trigonometry is based upon ratios of the sides of right triangles. The six trigonometric functions of a right triangle, with an acute angle ,are defined by ratios of two sides of the triangle. hyp opp adj The sides of the right triangle are:  the side opposite the acute angle ,  the side adjacent to the acute angle ,  and the hypotenuseof the right triangle.

  2. A A Right Triangle Trigonometry The hypotenuse is the longest side and is always opposite the right angle. The opposite and adjacent sides refer to another angle, other than the 90o.

  3. θ adj opp sin  = cos  = tan  = csc  = sec  = cot  = hyp adj hyp hyp adj opp adj opp Trigonometric Ratios hyp opp adj The trigonometric functions are: sine, cosine, tangent, cotangent, secant, and cosecant. S O HC A H T O A

  4. 1. 14 cm C 6 cm Finding an angle from a triangle To find a missing angle from a right-angled triangle we need to know two of the sides of the triangle. We can then choose the appropriate ratio, sin, cos or tan and use the calculator to identify the angle from the decimal value of the ratio. Find angle C • Identify/label the names of the sides. • b) Choose the ratio that contains BOTH of the letters.

  5. We have been given the adjacent and hypotenuse so we use COSINE: Cos A = Cos A = 1. Cos C = 14 cm C 6 cm h a Cos C = 0.4286 C = cos-1 (0.4286) C = 64.6o

  6. 2. Find angle x Given adj and opp need to use tan: Tan A = x 3 cm 8 cm Tan A = Tan x = a o Tan x = 2.6667 x = tan-1 (2.6667) x = 69.4o

  7. We have been given the adj and hyp so we use COSINE: Cos A = 3. 7 cm 30o k Cos A = Finding a side from a triangle Cos 30 = Cos 30 x 7 = k 6.1 cm = k

  8. 4. We have been given the opp and adj so we use TAN: Tan A = 50o 4 cm r Tan A = Tan 50 = Tan 50 x 4 = r 4.8 cm = r

  9. In a 45°-45°-90° triangle, the hypotenuse is √2 times as long as each leg. 45°-45°-90° Triangle Theorem 45° x√2 45° Hypotenuse = √2 * leg

  10. In a 30°-60°-90° triangle, the hypotenuse is twice as long as the shorter leg, and the longer leg is √3 times as long as the shorter leg. 30°-60°-90° Triangle Theorem 60° 30° x√3 Hypotenuse = 2 ∙ shorter leg Longer leg = √3 ∙ shorter leg

  11. Find the value of x By the Triangle Sum Theorem, the measure of the third angle is 45°. The triangle is a 45°-45°-90° right triangle, so the length x of the hypotenuse is √2 times the length of a leg. Ex. 1: Finding the hypotenuse in a 45°-45°-90° Triangle 3 3 45° x

  12. Hypotenuse = √2 ∙ leg x = √2 ∙ 3 x = 3√2 Ex. 1: Finding the hypotenuse in a 45°-45°-90° Triangle 3 3 45° x 45°-45°-90° Triangle Theorem Substitute values Simplify

  13. Find the values of s and t. Because the triangle is a 30°-60°-90° triangle, the longer leg is √3 times the length s of the shorter leg. Ex. 3: Finding side lengths in a 30°-60°-90° Triangle 60° 30°

  14. Statement: Longer leg = √3 ∙ shorter leg 5 = √3 ∙ s Reasons: 30°-60°-90° Triangle Theorem Ex. 3: Side lengths in a 30°-60°-90° Triangle 60° 30° Substitute values 5 √3s = Divide each side by √3 √3 √3 5 = s Simplify √3 Multiply numerator and denominator by √3 √3 5 = s √3 √3 5√3 Simplify = s 3

  15. Statement: Hypotenuse = 2 ∙ shorter leg Reasons: 30°-60°-90° Triangle Theorem The length t of the hypotenuse is twice the length s of the shorter leg. 60° 30° 5√3 t 2 ∙ Substitute values = 3 10√3 Simplify t = 3

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