Double Angle and Half Angle Formulas
Double angle and half angle formulas are trigonometric identities that express the sine, cosine, and tangent of double or half angles in terms of the sine and cosine of the original angle. These formulas are useful for simplifying trigonometric expressions and solving trigonometric equations. They are derived from the sum and difference identities for sine and cosine functions.
5 Key excerpts on "Double Angle and Half Angle Formulas"
- Lawrence S. Leff, Christina Pawlowski-Polanish(Authors)
- 2021(Publication Date)
- Barrons Educational Services(Publisher)
...You already know that cos 2 A = 1 – 2 sin 2 A. Solving for sin 2 A gives This formula expresses the sine function raised to the second power in terms of the cosine function raised to the first power. Power-reducing formulas for cos 2 A and tan 2 A can be obtained in a similar manner. POWER-REDUCING FORMULAS Lesson 11-5: Half-Angle Identities KEY IDEAS The graphs of and x do not coincide, indicating that. For example, since but. There are formulas, however, that allow a trigonometric function of a half angle to be expressed in terms of a trigonometric function of a single angle. FUNCTIONS OF A HALF ANGLE The half-angle formulas for sine, cosine, and tangent can be developed from the corresponding power-reducing formulas for these functions. For example, you learned in Lesson 11-4 that. Let 2 A = x ; then, which makes. Taking the square root of each side of the equation gives HALF-ANGLE FORMULAS The choice of a positive or negative sign in front of each radical depends on the sign of the trigonometric function in the quadrant in which lies. EXERCISE 1 Working with Half-Angle Formulas If and, find the values of and. SOLUTIONS First determine the quadrant in which lies. • It is given that or, equivalently, 270° ≤ x < 360°. To determine the quadrant in which lies, divide each member of the inequality by 2, which gives. Hence, lies in Quadrant II. • Because sine is positive in Quadrant II, use the positive value of the radical in the formula for, where cos : • Because cosine is negative in Quadrant II, use the negative value of the radical in the formula for, where : EXERCISE 2 Working with a Half-Angle Formula If ∠ A is obtuse and, what is the exact value of SOLUTION If sin, then Because ∠A is obtuse, cos A is negative. Thus,. Find tan using the half-angle formula for tangent where is a Quadrant I angle so tan is positive: HALF-ANGLE IDENTITIES The half-angle identities work whenever the two angles in the identity are in the ratio of 1 to 2...
eBook - ePub- Mike Aitken, Bill Broadhurst, Stephen Hladky(Authors)
- 2009(Publication Date)
- Garland Science(Publisher)
...The formulae that enable us to calculate the trigonometric functions are complicated but fortunately these have been programmed into scientific calculators or used to print tables. All we need to do is look up the values. A short table is provided in Appendix 3. The only catch is that with many tables we first need to convert the angle to degrees (multiply by 180/π), whereas with calculators we have to find out how to set it to accept angles in radians. More conveniently we can just type something like ‘= sin(2.73)’ into a spreadsheet program like Microsoft Excel ®. One note of caution: the word tangent has two meanings in geometry and trigonometry. One is the name of a trigonometric function, tan(α), the other refers to a line that just grazes a curve with the same slope or gradient as the curve at that point. These two meanings are closely related because the slope of a curve at a point is tan(α) where α is the angle between the tangent to the curve and the x axis. Box 4.3 Key values of sin(α) and cos(α) can be calculated from the definitions in terms of angles. The notation is the same as that used in Figure 4.10 together with the letter n, which represents any integer. When α = 0 or any multiple of 2π, x = 1, y = 0, r = 1 cos (0) = cos (2 n π) = x / r = 1 and sin (0) = sin (2 n π) = y / r = 0. Similarly when α = π/2 + 2 n π, x = 0, y = 1, r =. 1 cos (π/2 + 2 n π) = x / r = 0 and sin (π/2 + 2 n π) = y / r = 1, and continuing the pattern cos (π/2 + 2 n π) = − 1, sin (π + 2 n π) = 0, cos (3 π/2 + 2 n π) = 0, and sin (3 π/2 + 2 n π) = − 1. That. locates the peaks and troughs and the points where the curves in Figure 4.11 must cross the α axis. We can also find values for α = π/6, π/4, or π/3. For π/4 (45°), the defining triangle is an isosceles triangle with r = 1 and x = y. But then, by Pythagoras’ theorem r 2 = x 2 + y 2 = 2 x 2 x = y = r / 2 and sin (π/4) = cos (π/4) = 1 2. For π/6 we have to do a little more work as shown in Figure 1...
eBook - ePub- Adrian Waygood(Author)
- 2018(Publication Date)
- Routledge(Publisher)
...It would be nice to know where these terms came from, but their origins (well, for sine and cosine, anyway) are obscure and have been handed down to us via several ancient languages such as ancient Greek and Latin. So, it is far easier to simply accept them rather than to worry about where they came from or what they originally meant. The sine, cosine and tangent of any angle, θ, are defined in terms of the ratios of two of the triangle’s sides, as follows: sin ∠ θ = o p p o s i t e h y p o t e n u s e cos ∠ θ = a d j a c e n t h y p o t e n u n e tan ∠ θ = o p p o s i t e a d j a c e n t To help us remember these ratios, we can use the mnemonic,. ‘ SOH-CAH-TOA ’, as shown in Figure 24.4. Figure 24.4 We can determine the sine, cosine and tangent of any angle using a scientific calculator. The first step is to ensure that the calculator is set to measure degrees (rather than radians or grads), then hit the ‘ sin ’, ‘ cos ’ or ‘ tan ’ key, followed by the angle (some calculators may require you to enter the angle first). Although angles are traditionally measured in degrees, minutes and seconds, we would normally enter angles in decimal form – e.g. 50.5°, rather than as 50°30’. Refer to the calculator’s user manual for details. For sine and cosine, it’s useful (but not essential) to try to remember the following: angle sine cosine 0° 0.000 1.000 30° 0.500 0.866 60° 0.866 0.500 90° 1.000 0.000 As we will learn in the next chapter, the most commonly used of these, when it comes to working with phasor diagrams, is the cosine ratio. We can also use our scientific calculators to determine the angle if we know the value of its sine, cosine or tangent...
eBook - ePub- Alys Holden, Bronislaw Sammler, Bradley Powers, Steven Schmidt(Authors)
- 2015(Publication Date)
- Routledge(Publisher)
...angle. sin α = o p p o s i t e h y p o t e n u s e = o p p h y p = a c sin β = b c cos α = a d j a c e n t h y p o t e n u s e = a d j h y p = b c cos β = a c tan α = sin α cos α = o[--=PLGO-SEPARATO. R=--]p p o s i t e a d j a c e n t = o p p a d j = a b tan β = b a Figure H.4 Right triangle Notice that sinα = cosβ, as is always the case for complementary angles. Also notice that values for sine and cosine must always be less than 1 because the length of the hypotenuse is always larger than either of the two sides. When the ratio of the sides has been determined by finding the sine, cosine, or tangent, the corresponding angle is found by cross-referencing a trigonometric chart or using the “asin,” “acos,” or “atan” key or the 2nd function feature with the “sin,” “cos,” or “tan” key on a calculator. When trigonometric values for sine, cosine, or tangent are expressed, at least four decimal places are necessary to accurately define an angle. Find the sine, cosine, and tangent for both non-90° angles in the following triangle: sin α = sin 50° = 0.7660 cos α = cos 50° = 0.6428 tan α = tan 50° = 1.1918 α + β = 90° ⇒ β = 90° − 50° = 40° sin β = sin 40° = 0.6428 cos β = cos 40° = 0.7660 tan β = tan 40° = 0.8391 Figure H.5 Example 5 Find α and. β: sin α = o p p h y p = 9 15 = 0.6 α = sin − 1 (0.6) = 36.9 ° sin β = o p p h y p = 12 15 = 0.8 β = sin − 1 (0.8) = 53.1 ° α + β = 36.9 ° + 53.1 ° = 90 ° ✓ Figure H.6 Example 6 Two common right triangles are 45° triangles and 30°-60°-90° triangles. If the hypotenuse equals 1, the sides will equal the following: Figure H.7 45º and 30º-60º-90º triangles Triangles Like right triangles, any triangle is fully defined if three of its six parts are known and one of them is a side (2 sides and 1 angle, 2 angles and 1 side, or 3 sides) because the other three can then be calculated...
eBook - ePub- Surinder Virdi, Roy Baker, Narinder Kaur Virdi(Authors)
- 2014(Publication Date)
- Routledge(Publisher)
...CHAPTER 19 Trigonometry (2) Learning outcomes: (a) Show, that in ΔABC: (b) Apply the sine rule and the cosine rule to solve triangles and practical problems in construction (c) Find the area of a triangle if two sides and the included angle are given 19.19 The sine rule and the cosine rule In Chapter 13 we applied the trigonometrical ratios to determine the unknown angles and sides of right-angled triangles. For triangles without a right angle, the trigonometrical ratios cannot be applied directly; instead sine and cosine rules may be used to determine the unknown angles and sides. 19.1.1 The sine rule The sine rule states that in any triangle the ratio of the length of a side to the sine of the angle opposite that side is constant: where a, b and c are the sides of Δ ABC, as shown in Figure 19.1. Side a, b and c are opposite ∠ A, ∠ B and ∠ C, respectively. Figure 19.1 To prove the sine rule, draw a perpendicular from A to meet line BC at point D (Figure 19.2 a). Triangles ADB and ACD are right-angled triangles; therefore we can apply the trigonometric ratios to determine the length of AD. In Δ ADB, (1) (2) From equations (1) and (2): (3) In ΔBAC, draw a perpendicular from B to meet line CA at E, as shown in Figure 19.2b. Figure 19.2 (4) (5) From equations (4) and (5): (6) From equations (3) and (6): The above rule can be adapted if we have triangles PQR, XYZ and so on. In Δ PQR, In ΔXYZ, The sine rule may be used for the solution of triangles when: 1. two angles and one side are known 2. two sides and the angle opposite one of them are known. Example 19.1 In Δ ABC, ∠ B = 50°, ∠ C = 75° and AB = 80 cm. Find ∠ A and sides BC and AC. Solution: Δ ABC is shown in Figure 19.3. Figure 19.3 Example 19.2 The members of a roof truss slope at 35° and 60°, as shown in Figure 19.4...
